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A singular point of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s0855901.png" /> is an obstacle to the [[Analytic continuation|analytic continuation]] of an element of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s0855902.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s0855903.png" /> along any curve in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s0855904.png" />-plane.
+
{{TEX|done}}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s0855905.png" /> be defined by a Weierstrass element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s0855906.png" />, consisting of a power series
+
A singular point of an analytic function  $  f(z) $
 +
is an obstacle to the [[Analytic continuation|analytic continuation]] of an element of the function  $  f(z) $
 +
of a complex variable  $  z $
 +
along any curve in the  $  z $-
 +
plane.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s0855907.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
Let  $  f(z) $
 +
be defined by a Weierstrass element  $  (U( \zeta ,\  R),\  f _ \zeta  ) $,
 +
consisting of a power series $$ \tag{1}
 +
f _ \zeta    =   f _ \zeta  (z)  =
 +
\sum _{k=0} ^ \infty c _{k} (z - \zeta ) ^{k}  $$
 +
and its disc of convergence $$
 +
U( \zeta ,\  R)  =   \{ {z \in \overline{\mathbf C} } : {| z- \zeta | < R} \}
 +
$$
 +
with centre  $  \zeta \neq \infty $
 +
and radius of convergence  $  R > 0 $.
 +
Consider all possible curves  $  L: \  [0,\  1] \rightarrow \overline{\mathbf C}  $,
 +
i.e. continuous mappings  $  L: \  z = \phi (t) $
 +
of the interval  $  0 \leq t \leq 1 $
 +
into the extended complex plane  $  \overline{\mathbf C}  $,
 +
which begin at the centre of this element  $  \zeta $,
 +
$  \zeta = \phi (0) $.
 +
If the analytic continuation of the given element is possible along any such curve to any point  $  z \in \overline{\mathbf C}  $,
 +
then the complete analytic function  $  f(z) $
 +
thus obtained reduces to a constant: $  f(z) = \textrm const $.
 +
For non-trivial analytic functions  $  f(z) \neq \textrm const $,
 +
the existence of obstacles to the analytic continuation along certain curves  $  L $
 +
is characteristic.
  
and its disc of convergence
+
Let  $  a $
 +
be a point in the extended plane  $  \overline{\mathbf C}  $
 +
on a curve  $  L _{1} : \  z = \phi _{1} (t) $,
 +
$  a = \phi _{1} ( \tau _{1} ) $,
 +
$  0 < \tau _{1} \leq 1 $,
 +
$  \phi _{1} (0) = \zeta $,
 +
and on a curve  $  L _{2} : \  z = \phi _{2} (t) $,
 +
$  a = \phi _{2} ( \tau _{2} ) $,
 +
$  0 < \tau _{2} \leq 1 $,
 +
$  \phi _{2} (0) = \zeta $,
 +
and let analytic continuation along  $  L _{1} $
 +
and  $  L _{2} $
 +
to all preceding points  $  z = \phi _{1} (t) $,
 +
$  0 \leq t < \tau _{1} $,
 +
and  $  z = \phi _{2} (t) $,
 +
$  0 \leq t < \tau _{2} $,
 +
be possible. Two such curves  $  L _{1} $
 +
and  $  L _{2} $
 +
are said to be equivalent with respect to the analytic continuation of the given element  $  (U( \zeta ,\  R),\  f _ \zeta  ) $
 +
to the point  $  a $
 +
if there is for any neighbourhood  $  V(a) $
 +
of $  a $
 +
in  $  \overline{\mathbf C}  $
 +
a number  $  \epsilon > 0 $
 +
such that the Weierstrass element obtained from  $  (U( \zeta ,\  R),\  f _ \zeta  ) $
 +
by analytic continuation along  $  L _{1} $
 +
to any point  $  z ^ \prime  = \phi _{1} ( \tau ^ \prime  ) $,
 +
$  \tau _{1} - \epsilon < \tau ^ \prime  < \tau _{1} $,
 +
can be continued along a certain curve located in  $  V(a) $
 +
to an element obtained by continuation along  $  L _{2} $
 +
from  $  (U( \zeta ,\  r),\  f _ \zeta  ) $
 +
to any point  $  z = \phi _{2} ( \tau ^{\prime\prime} ) $,
 +
$  \tau _{2} - \epsilon < \tau ^{\prime\prime} < \tau _{2} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s0855908.png" /></td> </tr></table>
 
  
with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s0855909.png" /> and radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559010.png" />. Consider all possible curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559011.png" />, i.e. continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559012.png" /> of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559013.png" /> into the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559014.png" />, which begin at the centre of this element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559016.png" />. If the analytic continuation of the given element is possible along any such curve to any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559017.png" />, then the complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559018.png" /> thus obtained reduces to a constant: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559019.png" />. For non-trivial analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559020.png" />, the existence of obstacles to the analytic continuation along certain curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559021.png" /> is characteristic.
+
If analytic continuation to a point  $  a $
 +
is possible along a curve  $  L $,
 +
then it is also possible along all curves of the equivalence class  $  \{ L \} $
 +
containing  $  L $.  
 +
In this case, the pair  $  (a, \{ L \} ) $
 +
is said to be regular, or proper; it defines a single-valued regular branch of the analytic function $  f(z) $
 +
in a neighbourhood  $  V(a) $
 +
of the point.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559022.png" /> be a point in the extended plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559023.png" /> on a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559027.png" />, and on a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559031.png" />, and let analytic continuation along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559033.png" /> to all preceding points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559035.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559037.png" />, be possible. Two such curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559039.png" /> are said to be equivalent with respect to the analytic continuation of the given element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559040.png" /> to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559041.png" /> if there is for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559042.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559044.png" /> a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559045.png" /> such that the Weierstrass element obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559046.png" /> by analytic continuation along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559047.png" /> to any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559049.png" />, can be continued along a certain curve located in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559050.png" /> to an element obtained by continuation along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559051.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559052.png" /> to any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559054.png" />.
+
If analytic continuation along a curve $  L: \  z = \phi (t) $,  
 +
0 \leq t \leq 1 $,  
 +
$  \phi (0) = \zeta $,  
 +
which passes through  $  a $,  
 +
$  a = \phi ( \tau ) $,  
 +
0 < \tau \leq 1 $,  
 +
is possible to all points $  \phi (t) $,  
 +
0 \leq t < \tau $,  
 +
preceding  $  a $,  
 +
but is not possible to the point  $  a = \phi ( \tau ) $,
 +
then  $  a $
 +
is a singular point for analytic continuation of the element $  (U( \zeta ,\  R),\  f _ \zeta  ) $
 +
along the curve  $  L $.  
 +
In this instance it will also be singular for continuation along all curves of the equivalence class  $  \{ L \} $
 +
which pass through  $  a $.  
 +
The pair  $  (a, \{ L \} ) $,
 +
consisting of the point $  a \in \overline{\mathbf C}  $
 +
and the equivalence class  $  \{ L \} $
 +
of curves  $  L $
 +
which pass through  $  a $
 +
for each of which  $  a $
 +
is singular, is called a singular point of the analytic function  $  f(z) $
 +
defined by the element $  (U( \zeta ,\  R),\  f _ \zeta  ) $.  
 +
Two singular points  $  (a, \{ L \} ) $
 +
and  $  (b, \{ M \} ) $
 +
are said to coincide if  $  a = b $
 +
and if the classes  $  \{ L \} $
 +
and  $  \{ M \} $
 +
coincide. The point  $  a $
 +
of the extended complex plane  $  \overline{\mathbf C}  $
 +
is then called the projection, or  $  z $-
 +
coordinate, of the singular point $  (a, \{ L \} ) $;
 +
the singular point  $  (a, \{ L \} ) $
 +
is also said to lie above the point  $  a \in \mathbf C $.  
 +
In general, several (even a countable set of) different singular and regular pairs  $  (a, \{ L \} ) $
 +
obtained through analytic continuation of one and the same element  $  (U( \zeta ,\  R),\  f _ \zeta  ) $
 +
may lie above one and the same point  $  a \in \overline{\mathbf C}  $(
 +
cf. [[Branch point|Branch point]]).
  
If analytic continuation to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559055.png" /> is possible along a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559056.png" />, then it is also possible along all curves of the equivalence class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559057.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559058.png" />. In this case, the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559059.png" /> is said to be regular, or proper; it defines a single-valued regular branch of the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559060.png" /> in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559061.png" /> of the point.
+
If the radius of convergence of the initial series (1)  $  R < \infty $,
 +
then on the boundary circle  $  \Gamma = \{ {z \in \overline{\mathbf C} } : {| z- \zeta | = R} \} $
 +
of the disc of convergence  $  U( \zeta ,\  R) $
 +
there lies at least one singular point  $  a $
 +
of the element  $  (U( \zeta ,\  R),\  f _ \zeta  ) $,
 +
i.e. there is a singular point of the analytic function  $  f(z) $
 +
for continuation along the curves  $  z = \phi (t) $,
 +
0 \leq t \leq 1 $,  
 +
of the class $  \{ L \} $
 +
such that  $  z = \phi (t) \in U( \zeta ,\  R) $
 +
when  $  0 \leq t < 1 $,
 +
$  a = \phi (1) $.  
 +
In other words, a singular point of the element  $  (U( \zeta ,\  R),\  f _ \zeta  ) $
 +
is a point  $  a \in \Gamma $
 +
such that direct analytic continuation of the element  $  (U( \zeta ,\  R),\  f _ \zeta  ) $
 +
from the disc  $  U( \zeta ,\  R) $
 +
to any neighbourhood  $  V(a) $
 +
is impossible. In this situation, and generally in all cases where the lack of an obvious description of the class of curves  $  \{ L \} $
 +
cannot give rise to ambiguity, one usually restricts to the  $  z $-
 +
coordinate  $  a $
 +
of the singular point. The study of the position of singular points of an analytic function, in dependence on the properties of the sequence of coefficients  $  \{ c _{k} \} _{k=0} ^ \infty  $
 +
of the initial element  $  (U( \zeta ,\  R),\  f _ \zeta  ) $,
 +
is one of the main directions of research in function theory (see [[Hadamard theorem|Hadamard theorem]] on multiplication; [[Star of a function element|Star of a function element]], as well as [[#References|[1]]], [[#References|[3]]], [[#References|[5]]]). It is well-known, for example, that the singular points of the series $$
 +
f _{0} (z)  =   \sum _{k=0} ^ \infty b ^{k} z ^ {d ^ k} ,
 +
$$
 +
where  $  b \in \overline{\mathbf C}  $,
 +
$  | b | < 1 $,
 +
and  $  d \geq 2 $
 +
is a natural number, fill the whole boundary  $  \Gamma = \{ {z \in \overline{\mathbf C} } : {| z | = 1} \} $
 +
of its disc of convergence  $  U(0,\  1) $,
 +
although the sum of this series is continuous everywhere in the closed disc  $  \overline{U}  (0,\  1) = \{ {z \in \overline{\mathbf C} } : {| z | \leq 1} \} $.  
 +
Here,  $  \Gamma $
 +
is the natural boundary of the analytic function  $  f _{0} (z) $;
 +
analytic continuation of  $  f _{0} (z) $
 +
across the boundary of the disc  $  U(0,\  1) $
 +
is impossible.
  
If analytic continuation along a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559064.png" />, which passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559067.png" />, is possible to all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559069.png" />, preceding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559070.png" />, but is not possible to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559071.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559072.png" /> is a singular point for analytic continuation of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559073.png" /> along the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559074.png" />. In this instance it will also be singular for continuation along all curves of the equivalence class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559075.png" /> which pass through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559076.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559077.png" />, consisting of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559078.png" /> and the equivalence class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559079.png" /> of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559080.png" /> which pass through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559081.png" /> for each of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559082.png" /> is singular, is called a singular point of the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559083.png" /> defined by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559084.png" />. Two singular points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559086.png" /> are said to coincide if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559087.png" /> and if the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559089.png" /> coincide. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559090.png" /> of the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559091.png" /> is then called the projection, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559093.png" />-coordinate, of the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559094.png" />; the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559095.png" /> is also said to lie above the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559096.png" />. In general, several (even a countable set of) different singular and regular pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559097.png" /> obtained through analytic continuation of one and the same element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559098.png" /> may lie above one and the same point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s08559099.png" /> (cf. [[Branch point|Branch point]]).
+
Suppose that in a sufficiently small neighbourhood  $  V(a) = \{ {z \in \overline{\mathbf C} } : {| z-a | < R} \} $
 +
of a point  $  a \neq \infty $(
 +
or  $  V( \infty ) = \{ {z \in \overline{\mathbf C} } : {| z | > R} \} $),  
 +
analytic continuation along the curves of a specific class  $  \{ L \} $
 +
is possible to all points other than  $  a $,
 +
for all elements obtained, i.e. along all curves situated in the deleted neighbourhood  $  V ^ \prime  (a) = \{ {z \in \overline{\mathbf C} } : {0< | z-a | < R} \} $(
 +
respectively, $  V ^ \prime  ( \infty ) = \{ {z \in \mathbf C} : {| z-a | > R} \} $);
 +
the singular point $  (a, \{ L \} ) $
 +
is then called an [[Isolated singular point|isolated singular point]]. If analytic continuation of the elements obtained along the curves of the class $  \{ L \} $
 +
along all possible closed curves situated in  $  V ^ \prime  (a) $
 +
does not alter these elements, then the isolated singular point $  (a, \{ L \} ) $
 +
is called a single-valued singular point. This type of singular point can be a pole or an essential singular point: If an infinite limit  $  \mathop{\rm lim}\nolimits \  f(z) = \infty $
 +
exists when  $  z $
 +
moves towards  $  a $
 +
along the curves of the class  $  \{ L \} $,
 +
then the single-valued singular point  $  (a, \{ L \} ) $
 +
is called a [[Pole (of a function)|pole (of a function)]]; if no finite or infinite limit  $  \mathop{\rm lim}\nolimits \  f(z) $
 +
exists when  $  z $
 +
moves towards  $  a $
 +
along the curves of the class  $  \{ L \} $,
 +
then  $  (a, \{ L \} ) $
 +
is called an [[Essential singular point|essential singular point]]; the case of a finite limit corresponds to a regular point (a, \{ L \} ) $.  
 +
If analytic continuation of the elements obtained along the curves of the class  $  \{ L \} $
 +
along closed curves surrounding  $  a $
 +
in  $  V ^ \prime  (a) $
 +
alters these elements, then the isolated singular point  $  (a, \{ L \} ) $
 +
is called a [[Branch point|branch point]] or a many-valued singular point. The class of branch points is in turn subdivided into algebraic branch points and transcendental branch points (including logarithmic branch points, cf. [[Algebraic branch point|Algebraic branch point]]; [[Logarithmic branch point|Logarithmic branch point]]; [[Transcendental branch point|Transcendental branch point]]). If after a finite number  $  m \geq 2 $
 +
of single loops around  $  a $
 +
in the same direction within  $  V ^ \prime  (a) $,
 +
the elements obtained along the curves of the class  $  \{ L \} $
 +
take their original form, then  $  (a, \{ L \} ) $
 +
is an algebraic branch point and the number  $  m-1 $
 +
is called its order. Conversely, when the loops around  $  a $
 +
give more and more new elements,  $  (a, \{ L \} ) $
 +
is a transcendental branch point.
  
If the radius of convergence of the initial series (1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590100.png" />, then on the boundary circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590101.png" /> of the disc of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590102.png" /> there lies at least one singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590103.png" /> of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590104.png" />, i.e. there is a singular point of the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590105.png" /> for continuation along the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590107.png" />, of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590108.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590109.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590111.png" />. In other words, a singular point of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590112.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590113.png" /> such that direct analytic continuation of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590114.png" /> from the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590115.png" /> to any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590116.png" /> is impossible. In this situation, and generally in all cases where the lack of an obvious description of the class of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590117.png" /> cannot give rise to ambiguity, one usually restricts to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590118.png" />-coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590119.png" /> of the singular point. The study of the position of singular points of an analytic function, in dependence on the properties of the sequence of coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590120.png" /> of the initial element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590121.png" />, is one of the main directions of research in function theory (see [[Hadamard theorem|Hadamard theorem]] on multiplication; [[Star of a function element|Star of a function element]], as well as [[#References|[1]]], [[#References|[3]]], [[#References|[5]]]). It is well-known, for example, that the singular points of the series
+
For example, for the function $$
 +
f(z)  =  
 +
\frac{1}{(1+z ^{1/2} )(1 + z ^{1/6} )}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590122.png" /></td> </tr></table>
+
$$
 +
the points  $  a = 0 $,
 +
$  \infty $(
 +
for all curves) are algebraic branch points of order 5. As a point function,  $  f(z) $
 +
can be represented as a single-valued function only on the corresponding [[Riemann surface|Riemann surface]]  $  S $,
 +
consisting of 6 sheets over  $  \overline{\mathbf C}  $
 +
joined in a specific way above the points  $  0,\  \infty $.
 +
Moreover, three proper branches of  $  f(z) $
 +
lie above the point  $  a=1 $,
 +
which are single-valued on the three corresponding sheets of  $  S $;  
 +
on one sheet of  $  S $
 +
there is a pole of the second order, and on two sheets of  $  S $
 +
there are poles of the first order. In general, the introduction of the concept of a Riemann surface is particularly convenient and fruitful when studying the character of a singular point.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590124.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590125.png" /> is a natural number, fill the whole boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590126.png" /> of its disc of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590127.png" />, although the sum of this series is continuous everywhere in the closed disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590128.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590129.png" /> is the natural boundary of the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590130.png" />; analytic continuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590131.png" /> across the boundary of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590132.png" /> is impossible.
+
If the radius of convergence of the initial series (1)  $  R = \infty $,  
 +
then it represents an [[Entire function|entire function]]  $  f(z) $,  
 +
i.e. a function holomorphic in the entire finite plane  $  \mathbf C $.  
 +
When  $  f(z) \neq \textrm const $,  
 +
this function has a single isolated singular point  $  a = \infty $
 +
of single-valued character; if  $  a = \infty $
 +
is a pole, then  $  f(z) $
 +
is an entire rational function, or a polynomial; if  $  a = \infty $
 +
is an essential singular point, then  $  f(z) $
 +
is a transcendental entire function.
  
Suppose that in a sufficiently small neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590133.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590134.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590135.png" />), analytic continuation along the curves of a specific class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590136.png" /> is possible to all points other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590137.png" />, for all elements obtained, i.e. along all curves situated in the deleted neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590138.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590139.png" />); the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590140.png" /> is then called an [[Isolated singular point|isolated singular point]]. If analytic continuation of the elements obtained along the curves of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590141.png" /> along all possible closed curves situated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590142.png" /> does not alter these elements, then the isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590143.png" /> is called a single-valued singular point. This type of singular point can be a pole or an essential singular point: If an infinite limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590144.png" /> exists when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590145.png" /> moves towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590146.png" /> along the curves of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590147.png" />, then the single-valued singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590148.png" /> is called a [[Pole (of a function)|pole (of a function)]]; if no finite or infinite limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590149.png" /> exists when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590150.png" /> moves towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590151.png" /> along the curves of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590152.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590153.png" /> is called an [[Essential singular point|essential singular point]]; the case of a finite limit corresponds to a regular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590154.png" />. If analytic continuation of the elements obtained along the curves of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590155.png" /> along closed curves surrounding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590156.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590157.png" /> alters these elements, then the isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590158.png" /> is called a [[Branch point|branch point]] or a many-valued singular point. The class of branch points is in turn subdivided into algebraic branch points and transcendental branch points (including logarithmic branch points, cf. [[Algebraic branch point|Algebraic branch point]]; [[Logarithmic branch point|Logarithmic branch point]]; [[Transcendental branch point|Transcendental branch point]]). If after a finite number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590159.png" /> of single loops around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590160.png" /> in the same direction within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590161.png" />, the elements obtained along the curves of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590162.png" /> take their original form, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590163.png" /> is an algebraic branch point and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590164.png" /> is called its order. Conversely, when the loops around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590165.png" /> give more and more new elements, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590166.png" /> is a transcendental branch point.
+
A [[Meromorphic function|meromorphic function]]  $  f(z) $
 +
in the finite plane  $  \mathbf C $
 +
is obtained when analytic continuation of the series (1) leads to a single-valued analytic function  $  f(z) $
 +
in  $  \mathbf C $
 +
all singular points of which are poles. If  $  a = \infty $
 +
is a pole or a regular point, then the total number of poles of $  f(z) $
 +
in the extended plane  $  \overline{\mathbf C}  $
 +
is finite and  $  f(z) $
 +
is a rational function. For a transcendental meromorphic function $  f(z) $
 +
in  $  \mathbf C $,
 +
the point at infinity  $  a = \infty $
 +
can be a limit point of the poles — this is the simplest example of a non-isolated singular point of a single-valued analytic function. A meromorphic function in an arbitrary domain  $  D \subset \overline{\mathbf C}  $
 +
is defined in the same way.
  
For example, for the function
+
Generally speaking, the projections of non-isolated singular points can form different sets of points in the extended complex plane  $  \overline{\mathbf C}  $.
 +
In particular, whatever the domain  $  D \subset \overline{\mathbf C}  $,
 +
an analytic function  $  f _{D} (z) $
 +
exists in  $  D $
 +
for which  $  D $
 +
is its natural domain of existence, and the boundary  $  \Gamma = \partial D $
 +
is its natural boundary; thus, analytic continuation of the function  $  f _{D} (z) $
 +
across the boundary of  $  D $
 +
is impossible. Here, the natural boundary  $  \Gamma $
 +
consists of accessible and inaccessible points (see [[Limit elements|Limit elements]]). If a point  $  a \in \Gamma $
 +
is accessible along the curves of a class  $  \{ L \} $(
 +
there may be several of these classes), all situated in  $  D $
 +
except for the end point  $  a $,
 +
then only singular points of the function $  f _{D} (z) $
 +
can lie above  $  a $,
 +
since if this were not the case, analytic continuation of  $  f _{D} (z) $
 +
across the boundary of  $  D $
 +
through a part of  $  \Gamma $
 +
in a neighbourhood of  $  a $
 +
would be possible; the accessible points form a dense set on  $  \Gamma $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590167.png" /></td> </tr></table>
 
  
the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590168.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590169.png" /> (for all curves) are algebraic branch points of order 5. As a point function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590170.png" /> can be represented as a single-valued function only on the corresponding [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590171.png" />, consisting of 6 sheets over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590172.png" /> joined in a specific way above the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590173.png" />. Moreover, three proper branches of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590174.png" /> lie above the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590175.png" />, which are single-valued on the three corresponding sheets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590176.png" />; on one sheet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590177.png" /> there is a pole of the second order, and on two sheets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590178.png" /> there are poles of the first order. In general, the introduction of the concept of a Riemann surface is particularly convenient and fruitful when studying the character of a singular point.
+
The role of the defining element of an analytic function  $  f(z) $
 +
of several complex variables  $  z = (z _{1} \dots z _{n} ) $,
 +
$  n > 1 $,
 +
can be played by, for example, a Weierstrass element  $  (U ^{n} ( \zeta ,\  R),\  f _ \zeta  ) $
 +
in the form of a multiple power series $$ \tag{2}
 +
f _ \zeta    =   f _ \zeta  (z)  =
 +
$$
 +
$$
 +
=
 +
\sum _ {k _{1} \dots k _{n} =0} ^ \infty c _ {k _{1} \dots k _ n} (z _{1} - \zeta _{1} ) ^ {k _ 1} \dots (z _{n} - \zeta _{n} ) ^ {k _ n}
 +
$$
 +
and the polydisc of convergence of this series $$
 +
U ^{n} ( \zeta ,\  r)  =
 +
\{ {z \in \mathbf C ^ n} : {| z _{v} - \zeta _{v} | < R _{v} ,
 +
v = 1 \dots n} \}
 +
,
 +
$$
 +
with centre  $  \zeta = ( \zeta _{1} \dots \zeta _{n} ) \in \mathbf C ^{n} $
 +
and radius of convergence  $  R = \{ R _{1} > 0 \dots R _{n} > 0 \} $.  
 +
By taking in the process of analytic continuation of the element (2) along all possible curves  $  L: \  [0,\  1] \rightarrow \mathbf C ^{n} $,
 +
mappings of the interval  $  0 \leq t \leq 1 $
 +
into the complex space  $  \mathbf C ^{n} $
 +
as basis, a general definition of the singular points  $  (a, \{ L \} ) $,  
 +
$  a \in \mathbf C ^{n} $,
 +
of the function  $  f(z) $
 +
is obtained, which is formally completely analogous to the one mentioned above for the case  $  n=1 $.
  
If the radius of convergence of the initial series (1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590179.png" />, then it represents an [[Entire function|entire function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590180.png" />, i.e. a function holomorphic in the entire finite plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590181.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590182.png" />, this function has a single isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590183.png" /> of single-valued character; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590184.png" /> is a pole, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590185.png" /> is an entire rational function, or a polynomial; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590186.png" /> is an essential singular point, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590187.png" /> is a transcendental entire function.
 
  
A [[Meromorphic function|meromorphic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590188.png" /> in the finite plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590189.png" /> is obtained when analytic continuation of the series (1) leads to a single-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590190.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590191.png" /> all singular points of which are poles. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590192.png" /> is a pole or a regular point, then the total number of poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590193.png" /> in the extended plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590194.png" /> is finite and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590195.png" /> is a rational function. For a transcendental meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590196.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590197.png" />, the point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590198.png" /> can be a limit point of the poles — this is the simplest example of a non-isolated singular point of a single-valued analytic function. A meromorphic function in an arbitrary domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590199.png" /> is defined in the same way.
+
However, as a result of the overdeterminacy of the [[Cauchy-Riemann equations]] when  $  n > 1 $
 +
and the resulting "large power" of analytic continuation, the case  $  n > 1 $
 +
differs radically from the case  $  n=1 $.  
 +
In particular, for  $  n > 1 $
 +
there are domains  $  D \subset \mathbf C ^{n} $
 +
which cannot be natural domains of existence of any single-valued analytic or holomorphic function. In other words, on specific sections of the boundary  $  \partial D $
 +
of this domain there are no singular points of any holomorphic function  $  f(z) $
 +
defined in  $  D $,
 +
and analytic continuation is possible across them. For example, the Osgood–Brown theorem holds: If a compact set  $  K $
 +
is situated in a bounded domain  $  D \subset \mathbf C ^{n} $
 +
such that  $  D \setminus K $
 +
is also a domain, and if a function  $  f(z) $
 +
is holomorphic in  $  D \setminus K $,  
 +
then it can be holomorphically continued onto the whole domain  $  D $(
 +
see also [[Removable set|Removable set]]). The natural domains of existence of holomorphic functions are sometimes called domains of holomorphy (cf. [[Domain of holomorphy|Domain of holomorphy]]), and are characterized by specific geometric properties. Analytic continuation of a holomorphic function $  f(z) $
 +
which is originally defined in a domain  $  D \subset \mathbf C ^{n} $
 +
while retaining its single-valuedness makes it necessary to introduce, generally speaking, many-sheeted domains of holomorphy over  $  \mathbf C ^{n} $,
 +
or Riemann domains — analogues of Riemann surfaces (cf. [[Riemannian domain|Riemannian domain]]). In this interpretation, the singular points of a holomorphic function  $  f(z) $
 +
prove to be points of the boundary  $  \Gamma = \partial \widehat{D}  $
 +
of its domain of holomorphy  $  \widehat{D}  $.  
 +
The Osgood–Brown theorem shows that the connected components of  $  \Gamma $
 +
cannot form compact sets  $  K $
 +
such that the function $  f(z) $
 +
is holomorphic in $  \widehat{D}  \setminus K $.  
 +
In particular, for  $  n > 1 $
 +
there do not exist isolated singular points of holomorphic functions.
  
Generally speaking, the projections of non-isolated singular points can form different sets of points in the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590200.png" />. In particular, whatever the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590201.png" />, an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590202.png" /> exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590203.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590204.png" /> is its natural domain of existence, and the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590205.png" /> is its natural boundary; thus, analytic continuation of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590206.png" /> across the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590207.png" /> is impossible. Here, the natural boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590208.png" /> consists of accessible and inaccessible points (see [[Limit elements|Limit elements]]). If a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590209.png" /> is accessible along the curves of a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590210.png" /> (there may be several of these classes), all situated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590211.png" /> except for the end point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590212.png" />, then only singular points of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590213.png" /> can lie above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590214.png" />, since if this were not the case, analytic continuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590215.png" /> across the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590216.png" /> through a part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590217.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590218.png" /> would be possible; the accessible points form a dense set on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590219.png" />.
+
The simplest types of singular points of analytic functions of several complex variables are provided by meromorphic functions  $  f(z) $
 +
in a domain  $  D \subset \mathbf C ^{n} $,  
 +
$  n \geq 1 $,  
 +
which are characterized by the following properties: 1)  $  f(z) $
 +
is holomorphic everywhere in  $  D $
 +
with the exception of a [[Polar set|polar set]]  $  P $,  
 +
which consists of singular points; and 2) for any point  $  a \in P $
 +
there are a neighbourhood  $  V(a) $
 +
and a holomorphic function  $  \psi _{a} (z) $
 +
in  $  V(a) $
 +
such that the function  $  \phi _{a} (z) = \psi _{a} (z)f(z) $
 +
can be continued holomorphically to  $  V(a) $.  
 +
The singular points  $  a \in P $
 +
are then divided into poles, at which  $  \phi _{a} (a) \neq 0 $,  
 +
and points of indeterminacy, at which  $  \phi _{a} (a) = 0 $.  
 +
In the case of a pole,  $  \mathop{\rm lim}\nolimits \  f(z) = \infty $
 +
when  $  z $
 +
moves towards  $  a $,
 +
$  z \in D \setminus P $;
 +
in any neighbourhood of a point of indeterminacy,  $  f(z) $
 +
takes all values  $  w \in \mathbf C $.  
 +
For example, the meromorphic function  $  f(z) = z _{1} /z _{2} $
 +
in  $  \mathbf C ^{2} $
 +
has the straight line  $  P = \{ {z = (z _{1} ,\  z _{2} ) \in \mathbf C ^ 2} : {z _{2} = 0} \} $
 +
as its polar set; all points of this straight line are poles, with the exception of the point of indeterminacy  $  (0,\  0) $.  
 +
A meromorphic function  $  f(z) $
 +
in its domain of holomorphy  $  \widehat{D}  $
 +
can be represented globally in  $  \widehat{D}  $
 +
as the quotient of two holomorphic functions, i.e. its polar set  $  P $
 +
is an [[Analytic set|analytic set]].
  
The role of the defining element of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590220.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590221.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590222.png" />, can be played by, for example, a Weierstrass element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590223.png" /> in the form of a multiple power series
+
A point  $  a \in \mathbf C ^{n} $
 
+
is called a point of meromorphy of a function $  f(z) $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590224.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
if  $  f(z) $
 
+
is meromorphic in a certain neighbourhood of that point; thus, if a singular point is a point of meromorphy, then it is either a pole or a point of indeterminacy. All singular points of the analytic function $  f(z) $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590225.png" /></td> </tr></table>
+
which are not points of meromorphy are sometimes called essential singular points. These include, for example, the branch points of $  f(z) $,  
 
+
i.e. the branching points of its (many-sheeted) domain of holomorphy $  \widehat{D}  $.  
and the polydisc of convergence of this series
+
The dimension of the set of all singular points of a holomorphic function $  f(z) $
 
+
is, in general, equal to $  2n-1 $.  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590226.png" /></td> </tr></table>
+
Given certain extra restrictions on $  f(z) $
 
+
this set proves to be analytic (and, consequently, is of smaller dimension; see [[#References|[2]]]).
with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590227.png" /> and radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590228.png" />. By taking in the process of analytic continuation of the element (2) along all possible curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590229.png" />, mappings of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590230.png" /> into the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590231.png" /> as basis, a general definition of the singular points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590232.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590233.png" />, of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590234.png" /> is obtained, which is formally completely analogous to the one mentioned above for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590235.png" />.
 
 
 
However, as a result of the overdeterminacy of the [[Cauchy-Riemann equations]] when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590236.png" /> and the resulting "large power" of analytic continuation, the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590237.png" /> differs radically from the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590238.png" />. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590239.png" /> there are domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590240.png" /> which cannot be natural domains of existence of any single-valued analytic or holomorphic function. In other words, on specific sections of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590241.png" /> of this domain there are no singular points of any holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590242.png" /> defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590243.png" />, and analytic continuation is possible across them. For example, the Osgood–Brown theorem holds: If a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590244.png" /> is situated in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590245.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590246.png" /> is also a domain, and if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590247.png" /> is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590248.png" />, then it can be holomorphically continued onto the whole domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590249.png" /> (see also [[Removable set|Removable set]]). The natural domains of existence of holomorphic functions are sometimes called domains of holomorphy (cf. [[Domain of holomorphy|Domain of holomorphy]]), and are characterized by specific geometric properties. Analytic continuation of a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590250.png" /> which is originally defined in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590251.png" /> while retaining its single-valuedness makes it necessary to introduce, generally speaking, many-sheeted domains of holomorphy over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590252.png" />, or Riemann domains — analogues of Riemann surfaces (cf. [[Riemannian domain|Riemannian domain]]). In this interpretation, the singular points of a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590253.png" /> prove to be points of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590254.png" /> of its domain of holomorphy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590255.png" />. The Osgood–Brown theorem shows that the connected components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590256.png" /> cannot form compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590257.png" /> such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590258.png" /> is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590259.png" />. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590260.png" /> there do not exist isolated singular points of holomorphic functions.
 
 
 
The simplest types of singular points of analytic functions of several complex variables are provided by meromorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590261.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590262.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590263.png" />, which are characterized by the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590264.png" /> is holomorphic everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590265.png" /> with the exception of a [[Polar set|polar set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590266.png" />, which consists of singular points; and 2) for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590267.png" /> there are a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590268.png" /> and a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590269.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590270.png" /> such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590271.png" /> can be continued holomorphically to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590272.png" />. The singular points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590273.png" /> are then divided into poles, at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590274.png" />, and points of indeterminacy, at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590275.png" />. In the case of a pole, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590276.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590277.png" /> moves towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590278.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590279.png" />; in any neighbourhood of a point of indeterminacy, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590280.png" /> takes all values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590281.png" />. For example, the meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590282.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590283.png" /> has the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590284.png" /> as its polar set; all points of this straight line are poles, with the exception of the point of indeterminacy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590285.png" />. A meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590286.png" /> in its domain of holomorphy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590287.png" /> can be represented globally in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590288.png" /> as the quotient of two holomorphic functions, i.e. its polar set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590289.png" /> is an [[Analytic set|analytic set]].
 
 
 
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590290.png" /> is called a point of meromorphy of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590291.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590292.png" /> is meromorphic in a certain neighbourhood of that point; thus, if a singular point is a point of meromorphy, then it is either a pole or a point of indeterminacy. All singular points of the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590293.png" /> which are not points of meromorphy are sometimes called essential singular points. These include, for example, the branch points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590294.png" />, i.e. the branching points of its (many-sheeted) domain of holomorphy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590295.png" />. The dimension of the set of all singular points of a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590296.png" /> is, in general, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590297.png" />. Given certain extra restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590298.png" /> this set proves to be analytic (and, consequently, is of smaller dimension; see [[#References|[2]]]).
 
  
 
====References====
 
====References====
Line 61: Line 354:
  
 
====Comments====
 
====Comments====
Sometimes a point at which an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590299.png" /> is not defined but at which it can be defined so as to remain analytic is called a [[Removable singular point|removable singular point]].
+
Sometimes a point at which an analytic function $  f(z) $
 +
is not defined but at which it can be defined so as to remain analytic is called a [[Removable singular point|removable singular point]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590300.png" />, a famous theorem concerning values of an analytic function in a neighbourhood of an essential singular point is Picard's great theorem (cf. [[Picard theorem|Picard theorem]]). See [[Fabry theorem|Fabry theorem]] for restrictions on the coefficients and powers so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590301.png" /> has a singular point.
+
If $  n =1 $,  
 +
a famous theorem concerning values of an analytic function in a neighbourhood of an essential singular point is Picard's great theorem (cf. [[Picard theorem|Picard theorem]]). See [[Fabry theorem|Fabry theorem]] for restrictions on the coefficients and powers so that $  f(z) $
 +
has a singular point.
  
 
The Osgood–Brown theorem is also known as the Hartogs extension theorem.
 
The Osgood–Brown theorem is also known as the Hartogs extension theorem.
Line 70: Line 366:
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4 {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Cohn, "Conformal mapping on Riemann surfaces" , Dover, reprint (1980) {{MR|0594937}} {{ZBL|0493.30001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.B. Conway, "Functions of one complex variable" , Springer (1978) {{MR|0503901}} {{ZBL|0887.30003}} {{ZBL|0277.30001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 6 {{MR|0635928}} {{ZBL|0471.32008}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) {{MR|0847923}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) {{MR|1111477}} {{ZBL|0683.32002}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> R. Narasimhan, "Several complex variables" , Univ. Chicago Press (1971) {{MR|0342725}} {{ZBL|0223.32001}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R. Remmert, "Funktionentheorie" , '''1''' , Springer (1984) {{MR|0753290}} {{ZBL|0542.30001}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> B. Kaupp, "Holomorphic functions of several variables" , de Gruyter (1983) (Translated from German) {{MR|}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4 {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Cohn, "Conformal mapping on Riemann surfaces" , Dover, reprint (1980) {{MR|0594937}} {{ZBL|0493.30001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.B. Conway, "Functions of one complex variable" , Springer (1978) {{MR|0503901}} {{ZBL|0887.30003}} {{ZBL|0277.30001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 6 {{MR|0635928}} {{ZBL|0471.32008}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) {{MR|0847923}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) {{MR|1111477}} {{ZBL|0683.32002}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> R. Narasimhan, "Several complex variables" , Univ. Chicago Press (1971) {{MR|0342725}} {{ZBL|0223.32001}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R. Remmert, "Funktionentheorie" , '''1''' , Springer (1984) {{MR|0753290}} {{ZBL|0542.30001}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> B. Kaupp, "Holomorphic functions of several variables" , de Gruyter (1983) (Translated from German) {{MR|}} {{ZBL|}} </TD></TR></table>
  
A singular point, or singularity, of an algebraic variety is a point at which smoothness is violated. More accurately, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590302.png" /> be an [[Algebraic variety|algebraic variety]] or a [[Scheme|scheme]] of finite type over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590303.png" />. Then a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590304.png" /> is said to be singular if the corresponding local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590305.png" /> is not regular (regularity of a local Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590306.png" /> with maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590307.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590308.png" />). The set of singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590309.png" /> is closed in the [[Zariski topology|Zariski topology]] and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590310.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590311.png" /> is a reduced variety, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590312.png" /> is nowhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590313.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590314.png" /> is an isolated point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590315.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590316.png" /> is said to be an isolated singular point. In order to test whether or not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590317.png" /> is singular, Jacobi's criterion is used (see [[Smooth scheme|Smooth scheme]]).
+
A singular point, or singularity, of an algebraic variety is a point at which smoothness is violated. More accurately, let $  X $
 +
be an [[Algebraic variety|algebraic variety]] or a [[Scheme|scheme]] of finite type over a field $  k $.  
 +
Then a point $  x \in X $
 +
is said to be singular if the corresponding local ring $  {\mathcal O} _{X,x} $
 +
is not regular (regularity of a local Noetherian ring $  A $
 +
with maximal ideal $  \mathfrak m $
 +
means that $  \mathop{\rm dim}\nolimits \  \mathfrak m / \mathfrak m ^{2} = \mathop{\rm dim}\nolimits \  A $).  
 +
The set of singular points of $  X $
 +
is closed in the [[Zariski topology|Zariski topology]] and is denoted by $  \mathop{\rm Sing}\nolimits \  X $.  
 +
If $  X $
 +
is a reduced variety, then $  \mathop{\rm Sing}\nolimits \  X $
 +
is nowhere dense in $  X $.  
 +
If $  x $
 +
is an isolated point in $  \mathop{\rm Sing}\nolimits \  X $,  
 +
then $  x $
 +
is said to be an isolated singular point. In order to test whether or not $  x \in X $
 +
is singular, Jacobi's criterion is used (see [[Smooth scheme|Smooth scheme]]).
 +
 
 +
A proper birational morphism  $  \pi : \overline{X}  \rightarrow X $,
 +
where  $  \overline{X}  $
 +
is a smooth variety, is called a resolution of singularities (a desingularization) of the algebraic variety  $  X $.
 +
The existence of a resolution of singularities has been proved for a broad class of varieties, in particular, for all varieties over a field of characteristic  $  0 $(
 +
see [[#References|[13]]]). As a rule, it is not unique. A resolution of singularities is used to introduce various invariants of a variety  $  X $;
 +
an example of this are the cohomology spaces  $  H ^{i} ( \overline{X}  ,\  {\mathcal O} _ {\overline{X} } ) $.
 +
A normal variety  $  X $
 +
for which  $  H ^{i} ( \overline{X}  ,\  {\mathcal O} _ {\overline{X} } ) = 0 $
 +
for all  $  i > 0 $
 +
is called a variety with rational singularities. Toroidal singularities [[#References|[6]]] and singularities of Schubert varieties [[#References|[3]]] are rational. The dimension of the space  $  H ^{n-1} ( \overline{X}  ,\  {\mathcal O} _ {\overline{X} } ) $
 +
for an  $  n $-
 +
dimensional variety  $  X $
 +
is called the geometric genus of  $  X $.
 +
See also [[Resolution of singularities|Resolution of singularities]].
  
A proper birational morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590318.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590319.png" /> is a smooth variety, is called a resolution of singularities (a desingularization) of the algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590320.png" />. The existence of a resolution of singularities has been proved for a broad class of varieties, in particular, for all varieties over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590321.png" /> (see [[#References|[13]]]). As a rule, it is not unique. A resolution of singularities is used to introduce various invariants of a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590322.png" />; an example of this are the cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590323.png" />. A normal variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590324.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590325.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590326.png" /> is called a variety with rational singularities. Toroidal singularities [[#References|[6]]] and singularities of Schubert varieties [[#References|[3]]] are rational. The dimension of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590327.png" /> for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590328.png" />-dimensional variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590329.png" /> is called the geometric genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590330.png" />. See also [[Resolution of singularities|Resolution of singularities]].
+
The theory of deformations of singularities, i.e. of varieties with singular points, is constructed in parallel with the theory of deformations of (smooth) algebraic varieties. A [[Flat morphism|flat morphism]]  $  f: X \rightarrow S $
 +
for which  $  f ^ {\  -1} (s _{0} ) = X _{0} $
 +
for a certain  $  s _{0} \in S $
 +
is called a [[Deformation|deformation]] of $  X _{0} $;
 +
the space  $  S $
 +
is called the base of the deformation. For a variety $  X _{0} $
 +
with an isolated singular point there is a versal deformation which contains all deformations of the variety $  X _{0} $.  
 +
The singularity can be rigid, i.e. the base of the versal deformation consists of one point and all its deformations are trivial [[#References|[4]]]. The opposite of rigid singular points are smoothable singular points, in the base  $  S $
 +
of the versal deformation of which are points such that  $  X _{s} = f ^ {\  -1} (s) $
 +
are non-singular. The set  $  D $
 +
of points  $  s \in S $
 +
with singular  $  X _{s} $
 +
is called the discriminant subset.
  
The theory of deformations of singularities, i.e. of varieties with singular points, is constructed in parallel with the theory of deformations of (smooth) algebraic varieties. A [[Flat morphism|flat morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590331.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590332.png" /> for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590333.png" /> is called a [[Deformation|deformation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590334.png" />; the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590335.png" /> is called the base of the deformation. For a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590336.png" /> with an isolated singular point there is a versal deformation which contains all deformations of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590337.png" />. The singularity can be rigid, i.e. the base of the versal deformation consists of one point and all its deformations are trivial [[#References|[4]]]. The opposite of rigid singular points are smoothable singular points, in the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590338.png" /> of the versal deformation of which are points such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590339.png" /> are non-singular. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590340.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590341.png" /> with singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590342.png" /> is called the discriminant subset.
+
An important part in the study of deformations is played by the action of the monodromy group  $  \pi _{1} (S \setminus D) $
 +
on cohomology spaces of fibres of $  X $.
  
An important part in the study of deformations is played by the action of the monodromy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590343.png" /> on cohomology spaces of fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590344.png" />.
 
  
A proper morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590345.png" /> is called a simultaneous resolution of singularities of a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590346.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590347.png" /> is a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590348.png" />-scheme, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590349.png" /> the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590350.png" /> is a resolution of singularities. The versal deformation of simple singular points (see below) permits a simultaneous resolution after a certain finite covering of its base, whereby the Weyl group of the corresponding root system serves as the Galois group of the covering (see [[#References|[5]]]).
+
A proper morphism $  \pi : \  \overline{X}  \rightarrow X $
 +
is called a simultaneous resolution of singularities of a family $  X \rightarrow S $
 +
if $  \overline{X}  $
 +
is a smooth $  S $-
 +
scheme, and for any s \in S $
 +
the morphism $  \overline{X}  _{s} \rightarrow X _{s} $
 +
is a resolution of singularities. The versal deformation of simple singular points (see below) permits a simultaneous resolution after a certain finite covering of its base, whereby the Weyl group of the corresponding root system serves as the Galois group of the covering (see [[#References|[5]]]).
  
 
===Singular points of a complex hypersurface.===
 
===Singular points of a complex hypersurface.===
Let a hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590351.png" /> be defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590352.png" /> by one equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590353.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590354.png" /> is a polynomial (or germ of an analytic function at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590355.png" />). The ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590356.png" /> in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590357.png" /> is called the Jacobi ideal of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590358.png" />; the singular point 0 is isolated if and only if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590359.png" /> has finite dimension. The dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590360.png" /> of this space is called the Milnor number of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590361.png" />, and coincides with the rank of the free Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590362.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590363.png" /> is defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590364.png" />, for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590365.png" />. More precisely, the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590366.png" /> is homotopically equivalent to a bouquet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590367.png" /> spheres of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590368.png" /> (see [[#References|[12]]]). The base of the versal deformation of this singularity is non-singular and is also of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590369.png" /> (see [[#References|[9]]]). The simplest example is a non-degenerate quadratic singularity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590370.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590371.png" />.
+
Let a hypersurface $  X $
 +
be defined in $  \mathbf C ^{n+1} $
 +
by one equation $  f(x _{0} \dots x _{n} ) = 0 $,  
 +
where $  f $
 +
is a polynomial (or germ of an analytic function at the point 0 $).  
 +
The ideal $  J(f \  ) = ( \partial f/ \partial x _{0} \dots \partial f/ \partial x _{n} ) $
 +
in the ring $  \mathbf C \{ x _{0} \dots x _{n} \} $
 +
is called the Jacobi ideal of the polynomial $  f $;  
 +
the singular point 0 is isolated if and only if the space $  \mathbf C \{ x _{0} \dots x _{n} \} /J(f \  ) $
 +
has finite dimension. The dimension $  \mu $
 +
of this space is called the Milnor number of the polynomial $  f $,  
 +
and coincides with the rank of the free Abelian group $  H _{n} (X _ \epsilon  ,\  \mathbf Z ) $,  
 +
where $  X _ \epsilon  $
 +
is defined by the equation $  f(x _{0} \dots x _{n} ) = \epsilon $,  
 +
for small $  \epsilon \neq 0 $.  
 +
More precisely, the manifold $  X _ \epsilon  $
 +
is homotopically equivalent to a bouquet of $  \mu $
 +
spheres of dimension $  n $(
 +
see [[#References|[12]]]). The base of the versal deformation of this singularity is non-singular and is also of dimension $  \mu $(
 +
see [[#References|[9]]]). The simplest example is a non-degenerate quadratic singularity $  x _{0} ^{2} + \dots + x _{n} ^{2} = 0 $;  
 +
here $  \mu = 1 $.
 +
 
  
 
A singularity such that in a deformation only a finite number of other singularities appear is called a simple singular point of the hypersurface [[#References|[9]]]; the hypersurface is then defined by one of the following equations:
 
A singularity such that in a deformation only a finite number of other singularities appear is called a simple singular point of the hypersurface [[#References|[9]]]; the hypersurface is then defined by one of the following equations:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590372.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590373.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590374.png" />;
+
$  A _ \mu  : $:  
 +
$  x _{0} ^ {\mu + 1} + x _{1} ^{2} + \dots + x _{n} ^{2} = 0 $,
 +
$  \mu \geq 1 $;
 +
 
 +
 
 +
$  D _ \mu  $:  
 +
$  x _{0} ^ {\mu -1} + x _{0} x _{1} ^{2} + x _{2} ^{2} + \dots + x _{n} ^{2} = 0 $,
 +
$  \mu \geq 4 $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590375.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590376.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590377.png" />;
 
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590378.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590379.png" />;
+
$  E _{6} $:
 +
$  x _{0} ^{4} + x _{1} ^{3} + x _{2} ^{2} + \dots + x _{n} ^{2} = 0 $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590380.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590381.png" />;
 
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590382.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590383.png" />.
+
$  E _{7} $:
 +
$  x _{0} ^{3} x _{1} + x _{1} ^{3} + x _{2} ^{2} + \dots + x _{n} ^{2} = 0 $;
  
The lower index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590384.png" /> is the Milnor number of the singularity. In the case of surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590385.png" /> these singularities are called Du Val singularities, or rational singular double points. These singular points can also be characterized by the fact that the intersection form on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590386.png" /> is definite. Next, in order of complexity, the unimodal singularities are classified [[#References|[9]]]. Natural analogues of these concepts are studied, as well as their connection with catastrophe theory [[#References|[10]]]. Many theorems on the singular points of hypersurfaces are extended to the singular points of complete intersections.
 
  
===Singular points of curves.===
+
$  E _{8} $:
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590387.png" /> be the local ring of a singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590388.png" /> of a curve, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590389.png" /> be its normalization; a principal invariant of a singular point is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590390.png" />. For an irreducible curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590391.png" />, its arithmetic genus is equal to the geometric genus plus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590392.png" /> (summation over all singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590393.png" />). Hence, for a plane curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590394.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590395.png" /> is the Milnor number while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590396.png" /> is the number of branches of the curve at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590397.png" />.
+
$  x _{0} ^{5} + x _{1} ^{3} + x _{2} ^{2} + \dots + x _{n} ^{2} = 0 $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590398.png" /> be a plane irreducible curve with a singularity of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590399.png" /> at the point 0 (see [[Multiplicity of a singular point|Multiplicity of a singular point]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590400.png" /> then permits a parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590402.png" />, which is written in the form
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590403.png" /></td> </tr></table>
+
The lower index  $  \mu $
 +
is the Milnor number of the singularity. In the case of surfaces  $  (n=2) $
 +
these singularities are called Du Val singularities, or rational singular double points. These singular points can also be characterized by the fact that the intersection form on the space  $  H _{n} (X _ \epsilon  ,\  \mathbf R ) $
 +
is definite. Next, in order of complexity, the unimodal singularities are classified [[#References|[9]]]. Natural analogues of these concepts are studied, as well as their connection with catastrophe theory [[#References|[10]]]. Many theorems on the singular points of hypersurfaces are extended to the singular points of complete intersections.
  
(a Puiseux expansion). The numbers
+
===Singular points of curves.===
 +
Let  $  A $
 +
be the local ring of a singular point  $  x $
 +
of a curve, and let  $  \overline{A}  $
 +
be its normalization; a principal invariant of a singular point is  $  \delta _{x} =  \mathop{\rm dim}\nolimits \  \overline{A}  /A $.
 +
For an irreducible curve  $  X $,
 +
its arithmetic genus is equal to the geometric genus plus  $  \sum _{x} \delta _{x} $(
 +
summation over all singular points of  $  X $).
 +
Hence, for a plane curve  $  2 \delta _{x} = \mu + r - 1 $,
 +
where  $  \mu $
 +
is the Milnor number while  $  r $
 +
is the number of branches of the curve at the point  $  x $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590404.png" /></td> </tr></table>
 
  
are called the characteristic exponents of this expansion, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590405.png" /> is the first non-integer exponent in the Puiseux expansion, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590406.png" /> is the first exponent indivisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590407.png" />, etc. The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590408.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590409.png" />, is called the characteristic of the singularity. Plane one-dimensional singularities are topologically equivalent if and only if their characteristics coincide (see [[#References|[8]]]).
+
Let  $  X \subset \mathbf C ^{2} $
 +
be a plane irreducible curve with a singularity of multiplicity  $  n $
 +
at the point 0 (see [[Multiplicity of a singular point|Multiplicity of a singular point]]). $  X $
 +
then permits a parametrization  $  x = t ^{n} $,  
 +
$  y = \sum _{i\geqn} a _{i} t ^{i} $,  
 +
which is written in the form $$
 +
y  =  \sum _{i} a _{i} x ^{i/n}  $$(
 +
a Puiseux expansion). The numbers $$
  
===Singular points of surfaces.===
+
\frac{m _ 1}{n _ 1}
Among the resolutions of singularities of normal surfaces, the minimal resolutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590410.png" /> through which all remaining resolutions pass, are uniquely distinguished. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590411.png" /> is a singular point of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590412.png" />, then the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590413.png" /> is said to be exceptional. The weight graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590414.png" /> of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590415.png" /> is a combinatorial invariant of the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590416.png" />, and its vertices correspond to the irreducible components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590417.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590418.png" />; the intersection points of the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590419.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590420.png" /> are represented by the edges between the corresponding vertices; a weight equal to the genus of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590421.png" />, and sometimes even to the index of self-intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590422.png" />, is attributed to the vertex. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590423.png" /> of intersections of components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590424.png" /> is negative definite; the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590425.png" /> is connected. The smallest positive divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590426.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590427.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590428.png" /> is called the fundamental cycle of the singularity. It always exists and its arithmetic genus
+
  <  
 +
\frac{m _ 2}{n _{1} n _ 2}
 +
  < \dots <  
 +
\frac{m _ g}{n _{1} \dots n _ g}
 +
  =  
 +
\frac{m _ g}{n}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590429.png" /></td> </tr></table>
+
$$
 +
are called the characteristic exponents of this expansion, where  $  m _{1} /n _{1} $
 +
is the first non-integer exponent in the Puiseux expansion,  $  m _{2} /n _{1} n _{2} $
 +
is the first exponent indivisible by  $  1/n _{1} $,
 +
etc. The sequence  $  \{ n,\  \beta _{1} \dots \beta _{g} \} $,
 +
where  $  \beta _{v} = (m _{v} n)/(n _{1} \dots n _{v} ) $,
 +
is called the characteristic of the singularity. Plane one-dimensional singularities are topologically equivalent if and only if their characteristics coincide (see [[#References|[8]]]).
  
is non-negative. A singular point is rational if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590430.png" />; in this case its multiplicity equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590431.png" />, while the dimension of the tangent Zariski space is greater by one [[#References|[1]]]. Elliptic singularities (i.e. singular points with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590432.png" />) are also studied [[#References|[7]]].
+
===Singular points of surfaces.===
 +
Among the resolutions of singularities of normal surfaces, the minimal resolutions  $  \pi : \  \overline{X}  \rightarrow X $
 +
through which all remaining resolutions pass, are uniquely distinguished. If  $  x $
 +
is a singular point of a surface  $  X $,
 +
then the curve  $  A = \pi ^{-1} (x) $
 +
is said to be exceptional. The weight graph  $  \Gamma $
 +
of the curve  $  A $
 +
is a combinatorial invariant of the singular point  $  x $,
 +
and its vertices correspond to the irreducible components  $  A _{i} $
 +
of  $  A $;
 +
the intersection points of the components  $  A _{i} $
 +
and  $  A _{j} $
 +
are represented by the edges between the corresponding vertices; a weight equal to the genus of the curve  $  A _{i} $,
 +
and sometimes even to the index of self-intersection  $  (A _{i} ^{2} ) $,
 +
is attributed to the vertex. The matrix  $  \| (A _{i} ,\  A _{j} ) \| $
 +
of intersections of components of  $  A $
 +
is negative definite; the graph  $  \Gamma $
 +
is connected. The smallest positive divisor  $  Z = \sum r _{i} A _{i} $
 +
such that  $  (Z,\  A _{i} ) \leq 0 $
 +
for all  $  i $
 +
is called the fundamental cycle of the singularity. It always exists and its arithmetic genus $$
 +
p(Z)  =  1 -  \mathop{\rm dim}\nolimits \  H ^{0} (Z,\  {\mathcal O} _{Z} ) +  \mathop{\rm dim}\nolimits \  H ^{1} (Z,\
 +
{\mathcal O} _{Z} )
 +
$$
 +
is non-negative. A singular point is rational if and only if $  p(Z) = 0 $;  
 +
in this case its multiplicity equals $  -(Z ^{2} ) $,  
 +
while the dimension of the tangent Zariski space is greater by one [[#References|[1]]]. Elliptic singularities (i.e. singular points with $  p(Z) = 1 $)  
 +
are also studied [[#References|[7]]].
  
 
====References====
 
====References====
Line 123: Line 552:
  
 
====Comments====
 
====Comments====
Let the hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590433.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590434.png" /> be defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590435.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590436.png" /> be an isolated singular point. The fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590437.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590438.png" /> is a small disc of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590439.png" /> around zero, with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590440.png" /> is called the Milnor fibration.
+
Let the hypersurface $  X $
 +
in $  \mathbf C ^{n+1} $
 +
be defined by $  f(x _{0} \dots x _{n} ) = 0 $
 +
and let 0 $
 +
be an isolated singular point. The fibration $  f: \  \mathbf C ^{n+1} \rightarrow D( \epsilon ) $,  
 +
where $  D( \epsilon ) $
 +
is a small disc of radius $  \epsilon $
 +
around zero, with fibre $  X _ \epsilon  = \{ {(x _{0} \dots x _{n} )} : {f(x _{0} \dots x _{n} ) = \epsilon} \} $
 +
is called the Milnor fibration.
 +
 
 +
The ring  $  \mathbf C \{ x _{0} \dots x _{n} \} $
 +
is the ring of convergent power series in  $  x _{0} \dots x _{n} $.
 +
 
  
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590441.png" /> is the ring of convergent power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590442.png" />.
+
The invariant  $  \delta _{x} $
 +
of a singular point  $  x $
 +
on a curve intuitively counts the number of double points concentrated at  $  x $,
 +
[[#References|[a9]]].
  
The invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590443.png" /> of a singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590444.png" /> on a curve intuitively counts the number of double points concentrated at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590445.png" />, [[#References|[a9]]].
+
Let  $  f $
 +
be a polynomial with isolated critical point at  $  0 \in \mathbf C ^{n+1} $.  
 +
A Morsification of $  f $
 +
is a polynomial mapping  $  F: \  \mathbf C ^{n+2} \rightarrow \mathbf C $
 +
such that  $  F(z,\  0) = f(z) $(
 +
i.e. $  F $
 +
is a one-dimensional deformation of $  f \  $)
 +
and each  $  f _ \lambda  (z) = F(z,\  \lambda ) $
 +
has only non-degenerate critical points in a neighbourhood of  $  0 \in \mathbf C ^{n+1} $
 +
for small enough  $  \lambda $.  
 +
These always exist. Some examples (for  $  n =2 $)
 +
are as follows, [[#References|[a6]]], Sect. 7. Let $$
 +
\Phi _ \mu  (x,\  \lambda )  =
 +
$$
 +
$$
 +
 +
\left \{
 +
\begin{array}{ll}
 +
{(x + \lambda ) ^{2} \dots (x + k \lambda ) ^ 2}  &  \textrm{ if }  \mu = 2k,  \\
 +
{(x+ \lambda ) ^{2} \dots (x+k \lambda ) ^{2} (x+(k+1) \lambda )}  &  \textrm{ if }  \mu  =  2k+1 ,  \\
 +
\end{array}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590446.png" /> be a polynomial with isolated critical point at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590447.png" />. A Morsification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590448.png" /> is a polynomial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590449.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590450.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590451.png" /> is a one-dimensional deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590452.png" />) and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590453.png" /> has only non-degenerate critical points in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590454.png" /> for small enough <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590455.png" />. These always exist. Some examples (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590456.png" />) are as follows, [[#References|[a6]]], Sect. 7. Let
+
\right .$$
 +
A _ \mu  $:  
 +
$  f(x,\  y) = x ^ {\mu +1} -y ^{2} $,
 +
$  F(x,\  y,\  \lambda ) = \Phi _ {\mu +1} (x,\  \lambda ) - y ^{2} $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590457.png" /></td> </tr></table>
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590458.png" /></td> </tr></table>
+
$  D _ \mu  $:
 +
$  f(x,\  y) = x ^{m-1} -xy ^{2} = x(x ^{m-2} -y ^{2} ) $,
 +
$  F(x,\  y,\  \lambda ) = x \Phi _ {\mu -2} (x,\  \lambda ) -xy ^{2} $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590459.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590460.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590461.png" />;
 
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590462.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590463.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590464.png" />;
+
$  E _{6} $:  
 +
$  f(x,\  y) = x ^{3} + y ^{4} $,
 +
$  F(x,\  y,\  \lambda ) = (x- \mu )(x ^{2} - \lambda y ^{2} ) + y ^{4} $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590465.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590466.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590467.png" />;
 
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590468.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590469.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590470.png" />,
+
$  E _{7} $:  
 +
$  f(x,\  y) = x ^{3} + xy ^{3} $,
 +
$  F(x,\  y,\  \lambda ) = (x- \mu )(x ^{2} + y ^{3} + \lambda y ^{2} - 6 \lambda xy) $,
  
where in the last two examples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590471.png" /> is chosen small enough as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590472.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590473.png" /> cf. [[#References|[a6]]].
 
  
It is of course not an accident that the simple hypersurface singularities received the [[Dynkin diagram|Dynkin diagram]] labels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590474.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590475.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590476.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590477.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590478.png" />. In the examples given above the separatrix diagrams of the Morsifications are in fact the corresponding Dynkin diagrams. (But this is not necessarily the case for every Morsification.) Here the separatrix diagram of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590479.png" /> consists of the critical points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590480.png" /> for some fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590481.png" /> (and a number of lines joining these points (the vertices), where there is a line joining two critical points if and only if there is an integral curve of the gradient vector field
+
where in the last two examples  $  \mu $
 +
is chosen small enough as a function of $  \lambda $.
 +
For  $  E _{8} $
 +
cf. [[#References|[a6]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590482.png" /></td> </tr></table>
+
It is of course not an accident that the simple hypersurface singularities received the [[Dynkin diagram|Dynkin diagram]] labels  $  A _ \mu  $,
 +
$  D _ \mu  $,
 +
$  E _{6} $,
 +
$  E _{7} $,
 +
$  E _{8} $.  
 +
In the examples given above the separatrix diagrams of the Morsifications are in fact the corresponding Dynkin diagrams. (But this is not necessarily the case for every Morsification.) Here the separatrix diagram of  $  F(x,\  y,\  \lambda ) $
 +
consists of the critical points of  $  F(x,\  y,\  \lambda ) $
 +
for some fixed  $  \lambda \neq 0 $(
 +
and a number of lines joining these points (the vertices), where there is a line joining two critical points if and only if there is an integral curve of the gradient vector field $$
 +
\left (
 +
\frac{\partial F( x,\  y ,\  \lambda )}{\partial x}
 +
,\
  
 +
\frac{\partial F (x,\  y,\  \lambda )}{\partial y}
 +
\right )
 +
$$
 
joining these points).
 
joining these points).
  
Line 156: Line 642:
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Arnol'd, "Critical points of smooth functions" R. James (ed.) , ''Proc. Internat. Congress Mathematicians (Vancouver, 1974)'' , '''1''' , Canad. Math. Congress (1975) pp. 19–39 {{MR|}} {{ZBL|0343.58001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Brieskorn, "Singular elements of semisimple algebraic groups" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''2''' , Gauthier-Villars (1971) pp. 279–284 {{MR|437798}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Brieskorn, "Singularitäten" ''Jahresber. Deutsch. Math. Verein.'' , '''78''' (1976) pp. 93–112 {{MR|0430301}} {{ZBL|0334.32011}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.M. Gusein-Zade, "Dynkin diagrams for singularities of functions of two variables" ''Funct. Anal. Appl.'' , '''8''' : 4 (1974) pp. 295–300 ''Funkts. Anal. i Prilozh.'' , '''8''' : 4 (1974) pp. 23–30 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S.M. [S.M. Gusein-Zade] Husein-Zade, "The monodromy groups of isolated singularities of hypersurfaces" ''Russian Math. Surveys'' , '''32''' : 2 (1977) pp. 23–65 ''Uspekhi Mat. Nauk'' , '''32''' : 2 (1977) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Hazewinkel, W. Hesselink, D. Siersma, F.D. Veldkamp, "The ubiquity of Coxeter–Dynkin diagrams" ''Nieuw Archief voor Wiskunde'' , '''25''' (1977) pp. 257–307 {{MR|0491673}} {{ZBL|0377.20037}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> K. Gawedzki, "Conformal field theory" , ''Sem. Bourbaki 1988/89'' , '''Exp. 704''' , Soc. Math. France (1989) pp. 95–126 {{MR|1056302}} {{MR|1040570}} {{MR|1030458}} {{ZBL|0699.53086}} {{ZBL|0699.53085}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German) {{MR|0886476}} {{ZBL|0588.14019}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) pp. 65 {{MR|0103191}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> P.J. Slodowy, "Simple singularities and simple algebraic groups" , ''Lect. notes in math.'' , '''815''' , Springer (1980) {{MR|0584445}} {{ZBL|0441.14002}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> V.I. Arnol'd, "Singularities of caustics and wave fronts" , Kluwer (1990) {{MR|}} {{ZBL|0734.53001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Arnol'd, "Critical points of smooth functions" R. James (ed.) , ''Proc. Internat. Congress Mathematicians (Vancouver, 1974)'' , '''1''' , Canad. Math. Congress (1975) pp. 19–39 {{MR|}} {{ZBL|0343.58001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Brieskorn, "Singular elements of semisimple algebraic groups" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''2''' , Gauthier-Villars (1971) pp. 279–284 {{MR|437798}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Brieskorn, "Singularitäten" ''Jahresber. Deutsch. Math. Verein.'' , '''78''' (1976) pp. 93–112 {{MR|0430301}} {{ZBL|0334.32011}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.M. Gusein-Zade, "Dynkin diagrams for singularities of functions of two variables" ''Funct. Anal. Appl.'' , '''8''' : 4 (1974) pp. 295–300 ''Funkts. Anal. i Prilozh.'' , '''8''' : 4 (1974) pp. 23–30 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S.M. [S.M. Gusein-Zade] Husein-Zade, "The monodromy groups of isolated singularities of hypersurfaces" ''Russian Math. Surveys'' , '''32''' : 2 (1977) pp. 23–65 ''Uspekhi Mat. Nauk'' , '''32''' : 2 (1977) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Hazewinkel, W. Hesselink, D. Siersma, F.D. Veldkamp, "The ubiquity of Coxeter–Dynkin diagrams" ''Nieuw Archief voor Wiskunde'' , '''25''' (1977) pp. 257–307 {{MR|0491673}} {{ZBL|0377.20037}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> K. Gawedzki, "Conformal field theory" , ''Sem. Bourbaki 1988/89'' , '''Exp. 704''' , Soc. Math. France (1989) pp. 95–126 {{MR|1056302}} {{MR|1040570}} {{MR|1030458}} {{ZBL|0699.53086}} {{ZBL|0699.53085}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German) {{MR|0886476}} {{ZBL|0588.14019}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) pp. 65 {{MR|0103191}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> P.J. Slodowy, "Simple singularities and simple algebraic groups" , ''Lect. notes in math.'' , '''815''' , Springer (1980) {{MR|0584445}} {{ZBL|0441.14002}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> V.I. Arnol'd, "Singularities of caustics and wave fronts" , Kluwer (1990) {{MR|}} {{ZBL|0734.53001}} </TD></TR></table>
  
A singular point of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590483.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590484.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590485.png" />. A singular point is isolated if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590486.png" /> does not vanish at points other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590487.png" /> in a sufficiently small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590488.png" />. A singular point is non-degenerate if
+
A singular point of a vector field $  X $
 
+
is a point $  a $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590489.png" /></td> </tr></table>
+
for which $  X(a) = 0 $.  
 
+
A singular point is isolated if $  X $
 +
does not vanish at points other than $  a $
 +
in a sufficiently small neighbourhood of $  a $.  
 +
A singular point is non-degenerate if $$
 +
\mathop{\rm det}\nolimits \  \left \|
 +
\frac{\partial x ^ i}{\partial a ^ j}
 +
\
 +
\right \|  \neq  0.
 +
$$
 
A non-degenerate singular point is always isolated.
 
A non-degenerate singular point is always isolated.
  
Line 170: Line 664:
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Arnol'd, "Singularities of caustics and wave fronts" , Kluwer (1990) {{MR|}} {{ZBL|0734.53001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Arnol'd, "Singularities of caustics and wave fronts" , Kluwer (1990) {{MR|}} {{ZBL|0734.53001}} </TD></TR></table>
  
A singular point of a differential equation
+
A singular point of a differential equation $$ \tag{1}
 
+
X(x,\  y) \  dy  =   Y(x,\  y) \  dx
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590490.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$
 
+
is any point $  (x _{0} ,\  y _{0} ) \in G $
is any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590491.png" /> which satisfies the condition
+
which satisfies the condition $$ \tag{2}
 +
X(x _{0} ,\  y _{0} )  =   Y(x _{0} ,\  y _{0} )  =   0;
 +
$$
 +
here  $  X,\  Y: \  G \rightarrow \mathbf R $
 +
are continuous functions in a certain domain  $  G \subset \mathbf R ^{2} $.  
 +
The points of  $  G $
 +
which do not satisfy condition (2) are called the ordinary points of the equation (1). The point  $  (x _{0} ,\  y _{0} ) \in G $
 +
is sometimes also called a singular point of the equation (1) if condition (2) is not fulfilled but if the [[Cauchy problem|Cauchy problem]] for equation (1) with initial data  $  (x _{0} ,\  y _{0} ) $
 +
has more than one solution.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590492.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
Equation (1) is a particular case of a system of differential equations in symmetric form: $$ \tag{3}
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590493.png" /> are continuous functions in a certain domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590494.png" />. The points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590495.png" /> which do not satisfy condition (2) are called the ordinary points of the equation (1). The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590496.png" /> is sometimes also called a singular point of the equation (1) if condition (2) is not fulfilled but if the [[Cauchy problem|Cauchy problem]] for equation (1) with initial data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590497.png" /> has more than one solution.
+
\frac{dx _ 1}{X _{1} (x)}
 +
  = \dots =  
 +
\frac{dx _ n}{X _{n} (x)}
 +
,
 +
$$
 +
where  $  n \geq 2 $,
 +
$  x = (x _{1} \dots x _{n} ) $
 +
and the functions  $  X _{i} : \  G \rightarrow \mathbf R $,
 +
$  i = 1 \dots n $,
 +
are continuous in a domain $  G \subset \mathbf R ^{n} $.  
 +
A point $  x _{0} \in G $
 +
is called a singular point of the system (3) if $  X _{i} (x _{0} ) = 0 $,
 +
$  i = 1 \dots n $.  
 +
In the opposite case  $  x _{0} $
 +
is an ordinary point of this system.
  
Equation (1) is a particular case of a system of differential equations in symmetric form:
+
Let  $  H $
 +
be the set of singular points of the system (3) in the domain  $  G $.
 +
If  $  x _{0} \in G \setminus H $,
 +
then an index  $  i _{0} \in \{ 1 \dots n \} $
 +
and a neighbourhood  $  U $
 +
of the point  $  x _{0} $
 +
exist such that the system (3) can be represented in  $  U $
 +
in the normal form $$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590498.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\frac{dx _ i}{dx _ {i _ 0}}
 +
  =   f _{i} (x) , 
 +
f _{i} \in C (U), 
 +
i \neq i _{0} .
 +
$$
 +
Thus, the behaviour of the integral curves of the system (3) in a neighbourhood of an ordinary point is described by theorems of the general theory of ordinary differential equations. In particular, the following parallelizability theorem holds: If through every point  $  x _{0} $
 +
of the set  $  G \setminus H $
 +
passes a unique integral curve of the system (3), then every point of this set has a neighbourhood  $  V $
 +
such that the family of arcs of integral curves of the system (3) which fill  $  V $
 +
is homeomorphic (and if  $  X _{i} \in C ^{1} (G) $,
 +
$  i = 1 \dots n $,
 +
diffeomorphic) to a family of parallel straight lines.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590499.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590500.png" /> and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590502.png" />, are continuous in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590503.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590504.png" /> is called a singular point of the system (3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590506.png" />. In the opposite case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590507.png" /> is an ordinary point of this system.
+
If  $  x _{0} \in H $,  
 +
then no pair  $  (i _{0} ,\  U) $
 +
exists which possesses the above property, and the integral curves of the system (3) can form different configurations around  $  x _{0} $.  
 +
Thus, for the equation $$
 +
(ax + by) \  dy  =   (cx + ey) \  dx,
 +
$$
 +
where  $  a,\  b,\  c,\  e \in \mathbf R $,
 +
while the matrix $$
 +
A   =   \left \|
 +
\begin{array}{ll}
 +
a  & b  \\
 +
c  & e  \\
 +
\end{array}
 +
\right \|
 +
$$
 +
is non-degenerate, the position of integral curves in a neighbourhood of the point  $  (0,0) $
 +
can be of the type of a saddle, a node, a centre, or a focus. The same name is then also given to the point $  (0,\  0) $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590508.png" /> be the set of singular points of the system (3) in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590509.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590510.png" />, then an index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590511.png" /> and a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590512.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590513.png" /> exist such that the system (3) can be represented in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590514.png" /> in the normal form
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590515.png" /></td> </tr></table>
+
The system (3) can be seen as the result of the elimination of the time $  t $
 
+
from an [[Autonomous system|autonomous system]] of differential equations $$ \tag{4}
Thus, the behaviour of the integral curves of the system (3) in a neighbourhood of an ordinary point is described by theorems of the general theory of ordinary differential equations. In particular, the following parallelizability theorem holds: If through every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590516.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590517.png" /> passes a unique integral curve of the system (3), then every point of this set has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590518.png" /> such that the family of arcs of integral curves of the system (3) which fill <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590519.png" /> is homeomorphic (and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590520.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590521.png" />, diffeomorphic) to a family of parallel straight lines.
+
\dot{x}  =  X(x), 
 
+
x \in \mathbf R ^{n} , 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590522.png" />, then no pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590523.png" /> exists which possesses the above property, and the integral curves of the system (3) can form different configurations around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590524.png" />. Thus, for the equation
+
X = (X _{1} \dots X _{n} ).
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590525.png" /></td> </tr></table>
+
If (4) is a system of class ( $  C $,  
 
+
uniqueness) in $  G $,  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590526.png" />, while the matrix
+
i.e. $  X \in C (G) $,  
 
+
and a unique trajectory of the system passes through every point of the domain $  G $,  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590527.png" /></td> </tr></table>
+
then the points of the set $  H $
 
+
will be stationary points (cf. [[Equilibrium position|Equilibrium position]]) for this trajectory. These points are often called the singular points of this system, insofar as they are (by definition) singular points of the vector field $  X $.  
is non-degenerate, the position of integral curves in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590528.png" /> can be of the type of a saddle, a node, a centre, or a focus. The same name is then also given to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590529.png" />.
+
The integral curves of the system (3) situated in $  G \setminus H $
 
+
are trajectories of the system (4) other than the stationary positions.
The system (3) can be seen as the result of the elimination of the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590530.png" /> from an [[Autonomous system|autonomous system]] of differential equations
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590531.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
 
 
 
If (4) is a system of class (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590532.png" />, uniqueness) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590533.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590534.png" />, and a unique trajectory of the system passes through every point of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590535.png" />, then the points of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590536.png" /> will be stationary points (cf. [[Equilibrium position|Equilibrium position]]) for this trajectory. These points are often called the singular points of this system, insofar as they are (by definition) singular points of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590537.png" />. The integral curves of the system (3) situated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590538.png" /> are trajectories of the system (4) other than the stationary positions.
 
  
 
Thus, the problem of the behaviour of integral curves of the system (3) in a neighbourhood of a singular point and the problem of the positioning of the trajectories of the system (4) in a neighbourhood of an equilibrium position are equivalent. Research into these problems follows two main directions.
 
Thus, the problem of the behaviour of integral curves of the system (3) in a neighbourhood of a singular point and the problem of the positioning of the trajectories of the system (4) in a neighbourhood of an equilibrium position are equivalent. Research into these problems follows two main directions.
  
The first course, which has its origins in the work of H. Poincaré , aims to explain the possible topological types of how the trajectories of the system (4) are situated in a neighbourhood of an isolated stationary point (which can always be considered to coincide with the origin of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590539.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590540.png" />) and to discover the analytic criteria needed to distinguish them. The most complete results have been obtained for the case where the system (4) can be represented in the form
+
The first course, which has its origins in the work of H. Poincaré , aims to explain the possible topological types of how the trajectories of the system (4) are situated in a neighbourhood of an isolated stationary point (which can always be considered to coincide with the origin of the coordinates $  O $
 
+
$  (x = 0) $)
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590541.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
and to discover the analytic criteria needed to distinguish them. The most complete results have been obtained for the case where the system (4) can be represented in the form $$ \tag{5}
 
+
\dot{x}  =   Ax + f(x) ,
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590542.png" /> is a constant non-degenerate matrix and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590543.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590544.png" />. In this case the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590545.png" /> is said to be a simple, or non-degenerate, singular point of the system (4). The following Grobman–Hartman theorem has been established for the system (5): If the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590546.png" /> does not have purely imaginary eigenvalues, while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590547.png" />, then there is a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590548.png" /> of a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590549.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590550.png" /> onto a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590551.png" /> of the same point which transfers the trajectories of the system (5) to the trajectories of the linear system
+
$$
 
+
where $  A $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590552.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
is a constant non-degenerate matrix and $  f(x) = o( \| x \| ) $
 +
when $  \| x \| \rightarrow 0 $.  
 +
In this case the point $  O $
 +
is said to be a simple, or non-degenerate, singular point of the system (4). The following Grobman–Hartman theorem has been established for the system (5): If the matrix $  A $
 +
does not have purely imaginary eigenvalues, while the function $  f \in C ^{1} (G) $,  
 +
then there is a homeomorphism $  h $
 +
of a neighbourhood $  U $
 +
of the point $  O $
 +
onto a neighbourhood $  V $
 +
of the same point which transfers the trajectories of the system (5) to the trajectories of the linear system $$ \tag{6}
 +
\dot{x}  =   Ax.
 +
$$
 +
The homeomorphism  $  h: \  U \rightarrow V $
 +
which realizes a topological correspondence between the trajectories of the systems (5) and (6) is not a diffeomorphism, in general (nor can it be replaced by one).
  
The homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590553.png" /> which realizes a topological correspondence between the trajectories of the systems (5) and (6) is not a diffeomorphism, in general (nor can it be replaced by one).
+
Under the conditions of this theorem, the stationary point  $  O $
 +
of the system (5) is of the same topological type as the stationary point  $  O $
 +
of the system (6). In particular, for a system of the second order, it will be a saddle if the eigenvalues  $  \lambda _{1} ,\  \lambda _{2} $
 +
of the matrix  $  A $
 +
satisfy the condition  $  \lambda _{1} \lambda _{2} < 0 $,
 +
and a topological node (node or focus) if  $  \lambda _{1} \lambda _{2} > 0 $(
 +
given purely imaginary  $  \lambda _{1} ,\  \lambda _{2} $,
 +
the point  $  O $
 +
for the system (6) is a centre, while for the system (5) it is either a centre, a focus or a centre-focus, cf. [[Centro-focus|Centro-focus]]; [[Centre and focus problem|Centre and focus problem]]; [[Saddle node|Saddle node]]; [[Node|Node]]; [[Focus|Focus]]).
  
Under the conditions of this theorem, the stationary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590554.png" /> of the system (5) is of the same topological type as the stationary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590555.png" /> of the system (6). In particular, for a system of the second order, it will be a saddle if the eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590556.png" /> of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590557.png" /> satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590558.png" />, and a topological node (node or focus) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590559.png" /> (given purely imaginary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590560.png" />, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590561.png" /> for the system (6) is a centre, while for the system (5) it is either a centre, a focus or a centre-focus, cf. [[Centro-focus|Centro-focus]]; [[Centre and focus problem|Centre and focus problem]]; [[Saddle node|Saddle node]]; [[Node|Node]]; [[Focus|Focus]]).
+
If the matrix  $  A $
 +
has purely imaginary or zero eigenvalues, then there are, in general, no topological equivalences between the systems (5) and (6) in a neighbourhood of the point $  O $.  
 +
Under these conditions, the behaviour of the trajectories of the system (5) in a neighbourhood of the point  $  O $
 +
has been studied in great detail in those cases where the matrix  $  A $
 +
has at most two eigenvalues with zero real parts while the function  $  f $
 +
is analytic. In particular, for a system of the second order with a non-zero matrix  $  A $,  
 +
all possible topological types of positioning of trajectories in a neighbourhood of $  O $
 +
are clarified, and the coefficient criteria needed to distinguish between them have been given, up to the distinction between a centre and a focus [[#References|[9]]]. Here, apart from a saddle, topological node or centre, the point  $  O $
 +
can be a saddle with two separatrices, a saddle-node (a neighbourhood  $  U $
 +
of the point  $  O $
 +
is divided by three trajectories (separatrices) adjoining  $  O $
 +
into three sectors: two hyperbolic sectors, filled by trajectories which leave  $  U $
 +
at both ends, and one parabolic sector, filled by trajectories which leave  $  U $
 +
at one end, while the other approaches  $  O $)
 +
or a point with elliptical sector (a neighbourhood  $  U $
 +
of this point is divided into 4 sectors: one hyperbolic, two parabolic and one elliptic, filled by trajectories which approach  $  O $
 +
at both ends). For a system of the second order with a zero matrix  $  A $,  
 +
algorithms for the resolution of singularities have been worked out (see, for example, [[Frommer method|Frommer method]] or local methods in [[#References|[12]]]) which, with the aid of a finite number of steps of the resolution process, give a clarification of the topological type of the point  $  O $,
 +
accurate up to the solution of the problem of distinguishing between a centre and a focus. This problem (see [[Centre and focus problem|Centre and focus problem]]) arises for a system of the second order in the form (5) when the matrix  $  A $
 +
has purely imaginary eigenvalues, and can arise in the case of two zero eigenvalues of this matrix. It is solved for particular classes of such systems (see, e.g., [[#References|[14]]]).
  
If the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590562.png" /> has purely imaginary or zero eigenvalues, then there are, in general, no topological equivalences between the systems (5) and (6) in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590563.png" />. Under these conditions, the behaviour of the trajectories of the system (5) in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590564.png" /> has been studied in great detail in those cases where the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590565.png" /> has at most two eigenvalues with zero real parts while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590566.png" /> is analytic. In particular, for a system of the second order with a non-zero matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590567.png" />, all possible topological types of positioning of trajectories in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590568.png" /> are clarified, and the coefficient criteria needed to distinguish between them have been given, up to the distinction between a centre and a focus [[#References|[9]]]. Here, apart from a saddle, topological node or centre, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590569.png" /> can be a saddle with two separatrices, a saddle-node (a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590570.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590571.png" /> is divided by three trajectories (separatrices) adjoining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590572.png" /> into three sectors: two hyperbolic sectors, filled by trajectories which leave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590573.png" /> at both ends, and one parabolic sector, filled by trajectories which leave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590574.png" /> at one end, while the other approaches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590575.png" />) or a point with elliptical sector (a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590576.png" /> of this point is divided into 4 sectors: one hyperbolic, two parabolic and one elliptic, filled by trajectories which approach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590577.png" /> at both ends). For a system of the second order with a zero matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590578.png" />, algorithms for the resolution of singularities have been worked out (see, for example, [[Frommer method|Frommer method]] or local methods in [[#References|[12]]]) which, with the aid of a finite number of steps of the resolution process, give a clarification of the topological type of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590579.png" />, accurate up to the solution of the problem of distinguishing between a centre and a focus. This problem (see [[Centre and focus problem|Centre and focus problem]]) arises for a system of the second order in the form (5) when the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590580.png" /> has purely imaginary eigenvalues, and can arise in the case of two zero eigenvalues of this matrix. It is solved for particular classes of such systems (see, e.g., [[#References|[14]]]).
+
An important characteristic of the isolated stationary point $  O $
 
+
of the system (4) is its Poincaré index. For $  n=2 $
An important characteristic of the isolated stationary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590581.png" /> of the system (4) is its Poincaré index. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590582.png" /> it is defined as the rotation of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590583.png" /> around the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590584.png" /> (cf. [[Rotation of a vector field|Rotation of a vector field]]) along a circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590585.png" /> of a sufficiently small radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590586.png" /> in the positive direction, measured in units of a complete revolution. For example, the index of a simple saddle is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590587.png" />, the index of a node, focus or centre is equal to 1. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590588.png" /> is arbitrary, the index of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590589.png" /> is defined as the degree of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590590.png" /> (cf. [[Degree of a mapping|Degree of a mapping]]) of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590591.png" /> of a sufficiently small radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590592.png" /> onto itself, defined by the formula:
+
it is defined as the rotation of the vector field $  X $
 
+
around the point $  O $(
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590593.png" /></td> </tr></table>
+
cf. [[Rotation of a vector field|Rotation of a vector field]]) along a circle $  \| x \| = \rho $
 
+
of a sufficiently small radius $  \rho $
This course of research has led to the general [[Qualitative theory of differential equations|qualitative theory of differential equations]], while the emphasis of the research has shifted from local to global problems — the study of the behaviour of the trajectories of the system (4) in the entire domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590594.png" />, which is taken more and more often as a smooth manifold of some kind.
+
in the positive direction, measured in units of a complete revolution. For example, the index of a simple saddle is equal to $  -1 $,  
 +
the index of a node, focus or centre is equal to 1. When $  n $
 +
is arbitrary, the index of the point $  O $
 +
is defined as the degree of the mapping $  h $(
 +
cf. [[Degree of a mapping|Degree of a mapping]]) of the sphere $  \| x \| = \rho $
 +
of a sufficiently small radius $  \rho $
 +
onto itself, defined by the formula: $$
 +
h(x)  = 
 +
\frac{\rho X (x)}{\| X(x) \|}
 +
.
 +
$$
 +
This course of research has led to the general [[Qualitative theory of differential equations|qualitative theory of differential equations]], while the emphasis of the research has shifted from local to global problems — the study of the behaviour of the trajectories of the system (4) in the entire domain $  G $,  
 +
which is taken more and more often as a smooth manifold of some kind.
  
 
The other course of research, based on the work of A.M. Lyapunov [[#References|[2]]], deals with studies of stability of solutions of systems of the form (4) (especially of equilibrium positions), as well as of non-autonomous systems of differential equations. This research is one of the branches of the theory of stability of motion (see [[Stability theory|Stability theory]]).
 
The other course of research, based on the work of A.M. Lyapunov [[#References|[2]]], deals with studies of stability of solutions of systems of the form (4) (especially of equilibrium positions), as well as of non-autonomous systems of differential equations. This research is one of the branches of the theory of stability of motion (see [[Stability theory|Stability theory]]).
  
In complex analysis, the concept of a singular point is introduced for a differential equation
+
In complex analysis, the concept of a singular point is introduced for a differential equation $$ \tag{7}
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590595.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
 
 
 
and also for a system of differential equations
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590596.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
\frac{d ^{n} w}{dz ^ n}
 +
  =
 +
P \left ( z,\  w,\ 
 +
\frac{dw}{dz}
 +
\dots
 +
\frac{d ^{n-1} w}{dz ^ n-1}
 +
\right ) ,
 +
$$
 +
and also for a system of differential equations $$ \tag{8}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590597.png" /> is a complex variable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590598.png" /> is a rational function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590599.png" /> or in the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590600.png" /> of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590602.png" />, the coefficients of which are known analytic functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590603.png" />. Any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590604.png" /> of the complex plane which is a singular point of at least one of the coefficients of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590605.png" /> is said to be a singularity for equation (7) (for the system (8)) (see Singular point of an analytic function). Singular points of an equation or system, as a rule, are also singular for their solutions as analytic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590606.png" />. They are called fixed singular points (cf. [[Fixed singular point|Fixed singular point]]) of these solutions. Moreover, the solutions of equation (7) (system (8)) can have movable singular points (cf. [[Movable singular point|Movable singular point]]), the position of which is determined by the initial data of the solution. Studies on various classes of equations of the form (7), (8), aimed at clarifying the analytic nature of the solutions in a neighbourhood of the singular points of the equations, and at clarifying the presence of movable singular points of various types in the solutions of these equations, is the subject of the [[Analytic theory of differential equations|analytic theory of differential equations]].
+
\frac{dw}{dz}
 +
  =  P(z,\  w) ,
 +
$$
 +
where $  z $
 +
is a complex variable, $  P $
 +
is a rational function in $  w,\  w ^ \prime  \dots w ^{(n-1)} $
 +
or in the components $  w _{1} \dots w _{n} $
 +
of the vector $  w $,  
 +
$  n \geq 1 $,  
 +
the coefficients of which are known analytic functions of $  z $.  
 +
Any point $  z _{0} $
 +
of the complex plane which is a singular point of at least one of the coefficients of the function $  P $
 +
is said to be a singularity for equation (7) (for the system (8)) (see Singular point of an analytic function). Singular points of an equation or system, as a rule, are also singular for their solutions as analytic functions in $  z $.  
 +
They are called fixed singular points (cf. [[Fixed singular point|Fixed singular point]]) of these solutions. Moreover, the solutions of equation (7) (system (8)) can have movable singular points (cf. [[Movable singular point|Movable singular point]]), the position of which is determined by the initial data of the solution. Studies on various classes of equations of the form (7), (8), aimed at clarifying the analytic nature of the solutions in a neighbourhood of the singular points of the equations, and at clarifying the presence of movable singular points of various types in the solutions of these equations, is the subject of the [[Analytic theory of differential equations|analytic theory of differential equations]].
  
 
====References====
 
====References====
Line 253: Line 866:
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.A. Krasnosel'skii, A.I. [A.I. Perov] Perow, A.I. [A.I. Povoloskii] Powolzki, P.P. [P.P. Zabreiko] Sabrejko, "Vektorfelder in der Ebene" , Akademie Verlag (1966) (Translated from Russian) {{MR|202121}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.A. Krasnosel'skii, A.I. [A.I. Perov] Perow, A.I. [A.I. Povoloskii] Powolzki, P.P. [P.P. Zabreiko] Sabrejko, "Vektorfelder in der Ebene" , Akademie Verlag (1966) (Translated from Russian) {{MR|202121}} {{ZBL|}} </TD></TR></table>
  
A singular point of a differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590607.png" /> is a point which is simultaneously irregular (critical) and improper for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590608.png" />. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590610.png" /> be two differentiable manifolds of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590611.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590612.png" />, respectively, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590613.png" /> be a differentiable mapping of the first onto the second, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590614.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590615.png" /> be local coordinates in them. If the rank of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590616.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590617.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590618.png" />, then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590619.png" /> is said to be regular at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590620.png" />. If the rank of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590621.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590622.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590623.png" />, then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590624.png" /> is said to be proper at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590625.png" />. At a singular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590626.png" />, the rank of this matrix is not equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590627.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590628.png" />. See also [[Singularities of differentiable mappings|Singularities of differentiable mappings]].
+
A singular point of a differentiable mapping $  f $
 +
is a point which is simultaneously irregular (critical) and improper for $  f $.  
 +
More precisely, let $  M ^{m} $
 +
and $  N ^{n} $
 +
be two differentiable manifolds of dimensions $  m $
 +
and $  n $,  
 +
respectively, let $  f: \  M ^{m} \rightarrow N ^{n} $
 +
be a differentiable mapping of the first onto the second, and let $  x ^{j} $
 +
and $  y ^{i} = f(x ^{j} ) $
 +
be local coordinates in them. If the rank of the matrix $  \| \partial y ^{i} / \partial x ^{j} \| $
 +
at a point $  a \in M ^{m} $
 +
is equal to $  m $,  
 +
then the mapping $  f $
 +
is said to be regular at $  a $.  
 +
If the rank of the matrix $  \| \partial y ^{i} / \partial x ^{j} \| $
 +
is equal to $  n $
 +
at a point $  a \in M ^{m} $,  
 +
then the mapping $  f $
 +
is said to be proper at $  a $.  
 +
At a singular point of $  f $,  
 +
the rank of this matrix is not equal to $  m $
 +
or $  n $.  
 +
See also [[Singularities of differentiable mappings|Singularities of differentiable mappings]].
  
 
''M.I. Voitsekhovskii''
 
''M.I. Voitsekhovskii''
  
A singular point of a real curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590629.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590630.png" /> at which the first partial derivatives vanish: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590631.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590632.png" />. A singular point is called a double point if at least one of the second partial derivatives of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590633.png" /> does not vanish. In studies on the structure of a curve in a neighbourhood of a singular point, the sign of the expression
+
A singular point of a real curve $  F(x,\  y) = 0 $
 
+
is a point $  (x _{0} ,\  y _{0} ) $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590634.png" /></td> </tr></table>
+
at which the first partial derivatives vanish: $  (F _{x} ^ {\  \prime} ) _{0} = 0 $,  
 
+
$  (F _{y} ^ {\  \prime} ) _{0} = 0 $.  
is studied. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590635.png" />, then the singular point is an isolated point (Fig.a); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590636.png" />, it is a node (or point of self-intersection) (Fig.b); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590637.png" />, then it is either an isolated point or is characterized by the fact that different branches of the curve have a common tangent at this point. If the branches of the curve are situated on different sides of the common tangent and on the same side of the common normal, then the singular point is called a cusp of the first kind (Fig.c); if the branches of the curve are situated on the same side of the common tangent and on the same side of the common normal, then the singular point is called a cusp of the second kind (Fig.d); if the branches are situated on different sides of the common normal and on different sides of the common tangent (Fig.e), or on the same side of the common tangent and on different sides of the common normal (Fig.f), then the singular point is called a point of osculation. See also [[Double point|Double point]].
+
A singular point is called a double point if at least one of the second partial derivatives of the function $  F(x,\  y) $
 +
does not vanish. In studies on the structure of a curve in a neighbourhood of a singular point, the sign of the expression $$
 +
\Delta  =   (F _{xx} ^ {\  \prime\prime} ) _{0} (F _{yy} ^ {\  \prime\prime} ) _{0} - (F _{xy} ^ {\  \prime\prime} ) _{0} ^{2}  $$
 +
is studied. If $  \Delta > 0 $,  
 +
then the singular point is an isolated point (Fig.a); if $  \Delta < 0 $,  
 +
it is a node (or point of self-intersection) (Fig.b); if $  \Delta = 0 $,  
 +
then it is either an isolated point or is characterized by the fact that different branches of the curve have a common tangent at this point. If the branches of the curve are situated on different sides of the common tangent and on the same side of the common normal, then the singular point is called a cusp of the first kind (Fig.c); if the branches of the curve are situated on the same side of the common tangent and on the same side of the common normal, then the singular point is called a cusp of the second kind (Fig.d); if the branches are situated on different sides of the common normal and on different sides of the common tangent (Fig.e), or on the same side of the common tangent and on different sides of the common normal (Fig.f), then the singular point is called a point of osculation. See also [[Double point|Double point]].
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s085590a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s085590a.gif" />
Line 287: Line 928:
 
Figure: s085590f
 
Figure: s085590f
  
If all partial derivatives of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590638.png" /> up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590639.png" /> inclusive vanish at a certain point and at least one of the derivatives of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590640.png" /> differs from zero, then this point is called a singular point of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590642.png" /> (a multiple point).
+
If all partial derivatives of the function $  F(x,\  y) $
 +
up to order $  (k-1) $
 +
inclusive vanish at a certain point and at least one of the derivatives of order $  k $
 +
differs from zero, then this point is called a singular point of order $  k $(
 +
a multiple point).
  
 
Points which differ in any of their properties from other points of the curve are sometimes called singular points; see, for example, [[Point of inflection|Point of inflection]]; [[Point of cessation|Point of cessation]]; [[Breaking point|Breaking point]]; [[Point of rectification|Point of rectification]]; [[Flat point|Flat point]].
 
Points which differ in any of their properties from other points of the curve are sometimes called singular points; see, for example, [[Point of inflection|Point of inflection]]; [[Point of cessation|Point of cessation]]; [[Breaking point|Breaking point]]; [[Point of rectification|Point of rectification]]; [[Flat point|Flat point]].
  
A singular point of a spatial curve defined by the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590643.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590644.png" /> is a point in a neighbourhood of which the rank of the matrix
+
A singular point of a spatial curve defined by the equations $  F(x,\  y,\  z) = 0 $,
 
+
$  G(x,\  y,\  z) = 0 $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590645.png" /></td> </tr></table>
+
is a point in a neighbourhood of which the rank of the matrix $$
 
+
\left \|
 +
\begin{array}{lll}
 +
F _{x} ^ {\  \prime}  &F _{y} ^ {\  \prime}  &F _{z} ^ {\  \prime}  \\
 +
G _{x} ^ \prime  &G _{y} ^ \prime  &G _{z} ^ \prime  \\
 +
\end{array}
 +
\right \|
 +
$$
 
is less than two.
 
is less than two.
  
Line 302: Line 953:
 
''A.B. Ivanov''
 
''A.B. Ivanov''
  
A singular point of a real surface is a point of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590646.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590647.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590648.png" /> at which the rank of the matrix
+
A singular point of a real surface is a point of the surface $  x = x(u,\  v) $,
 
+
$  y= y(u,\  v) $,  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590649.png" /></td> </tr></table>
+
$  z = z(u,\  v) $
 
+
at which the rank of the matrix $$
is less than two. If the surface is defined as the set of points whose coordinates satisfy an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590650.png" />, then a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590651.png" /> of the surface at which the first partial derivatives of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590652.png" /> vanish is called a singular point:
+
\left \|
 
+
\begin{array}{lll}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590653.png" /></td> </tr></table>
+
x _{u}  &y _{u}  &z _{u}  \\
 
+
x _{v}  &y _{v}  &z _{v}  \\
If not all second partial derivatives of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590654.png" /> vanish at the singular point, then the tangents of the surface at the singular point form a cone. If the tangent cone is non-degenerate, then the singular point is called a conic point; if the cone degenerates to two real planes, then the singular point is called a point of self-intersection of the surface; if the cone is imaginary, then the singular point is an isolated point of the surface.
+
\end{array}
 +
\right \|
 +
$$
 +
is less than two. If the surface is defined as the set of points whose coordinates satisfy an equation $  F(x,\  y,\  z) = 0 $,  
 +
then a point $  (x _{0} ,\  y _{0} ,\  z _{0} ) $
 +
of the surface at which the first partial derivatives of the function $  F(x,\  y,\  z) $
 +
vanish is called a singular point: $$
 +
(F _{x} ^ {\  \prime} ) _{0}  =  0, 
 +
(F _{y} ^ {\  \prime} ) _{0}  =   0, 
 +
(F _{z} ^ {\  \prime} ) _{0}  =   0.
 +
$$
 +
If not all second partial derivatives of the function $  F(x,\  y,\  z) $
 +
vanish at the singular point, then the tangents of the surface at the singular point form a cone. If the tangent cone is non-degenerate, then the singular point is called a conic point; if the cone degenerates to two real planes, then the singular point is called a point of self-intersection of the surface; if the cone is imaginary, then the singular point is an isolated point of the surface.
  
 
Singular points can form so-called singular curves of a surface: an [[Edge of regression|edge of regression]], lines of self-intersection, lines of osculation, and others.
 
Singular points can form so-called singular curves of a surface: an [[Edge of regression|edge of regression]], lines of self-intersection, lines of osculation, and others.

Latest revision as of 00:24, 22 December 2019


A singular point of an analytic function $ f(z) $ is an obstacle to the analytic continuation of an element of the function $ f(z) $ of a complex variable $ z $ along any curve in the $ z $- plane.

Let $ f(z) $ be defined by a Weierstrass element $ (U( \zeta ,\ R),\ f _ \zeta ) $, consisting of a power series $$ \tag{1} f _ \zeta = f _ \zeta (z) = \sum _{k=0} ^ \infty c _{k} (z - \zeta ) ^{k} $$ and its disc of convergence $$ U( \zeta ,\ R) = \{ {z \in \overline{\mathbf C} } : {| z- \zeta | < R} \} $$ with centre $ \zeta \neq \infty $ and radius of convergence $ R > 0 $. Consider all possible curves $ L: \ [0,\ 1] \rightarrow \overline{\mathbf C} $, i.e. continuous mappings $ L: \ z = \phi (t) $ of the interval $ 0 \leq t \leq 1 $ into the extended complex plane $ \overline{\mathbf C} $, which begin at the centre of this element $ \zeta $, $ \zeta = \phi (0) $. If the analytic continuation of the given element is possible along any such curve to any point $ z \in \overline{\mathbf C} $, then the complete analytic function $ f(z) $ thus obtained reduces to a constant: $ f(z) = \textrm const $. For non-trivial analytic functions $ f(z) \neq \textrm const $, the existence of obstacles to the analytic continuation along certain curves $ L $ is characteristic.

Let $ a $ be a point in the extended plane $ \overline{\mathbf C} $ on a curve $ L _{1} : \ z = \phi _{1} (t) $, $ a = \phi _{1} ( \tau _{1} ) $, $ 0 < \tau _{1} \leq 1 $, $ \phi _{1} (0) = \zeta $, and on a curve $ L _{2} : \ z = \phi _{2} (t) $, $ a = \phi _{2} ( \tau _{2} ) $, $ 0 < \tau _{2} \leq 1 $, $ \phi _{2} (0) = \zeta $, and let analytic continuation along $ L _{1} $ and $ L _{2} $ to all preceding points $ z = \phi _{1} (t) $, $ 0 \leq t < \tau _{1} $, and $ z = \phi _{2} (t) $, $ 0 \leq t < \tau _{2} $, be possible. Two such curves $ L _{1} $ and $ L _{2} $ are said to be equivalent with respect to the analytic continuation of the given element $ (U( \zeta ,\ R),\ f _ \zeta ) $ to the point $ a $ if there is for any neighbourhood $ V(a) $ of $ a $ in $ \overline{\mathbf C} $ a number $ \epsilon > 0 $ such that the Weierstrass element obtained from $ (U( \zeta ,\ R),\ f _ \zeta ) $ by analytic continuation along $ L _{1} $ to any point $ z ^ \prime = \phi _{1} ( \tau ^ \prime ) $, $ \tau _{1} - \epsilon < \tau ^ \prime < \tau _{1} $, can be continued along a certain curve located in $ V(a) $ to an element obtained by continuation along $ L _{2} $ from $ (U( \zeta ,\ r),\ f _ \zeta ) $ to any point $ z = \phi _{2} ( \tau ^{\prime\prime} ) $, $ \tau _{2} - \epsilon < \tau ^{\prime\prime} < \tau _{2} $.


If analytic continuation to a point $ a $ is possible along a curve $ L $, then it is also possible along all curves of the equivalence class $ \{ L \} $ containing $ L $. In this case, the pair $ (a, \{ L \} ) $ is said to be regular, or proper; it defines a single-valued regular branch of the analytic function $ f(z) $ in a neighbourhood $ V(a) $ of the point.

If analytic continuation along a curve $ L: \ z = \phi (t) $, $ 0 \leq t \leq 1 $, $ \phi (0) = \zeta $, which passes through $ a $, $ a = \phi ( \tau ) $, $ 0 < \tau \leq 1 $, is possible to all points $ \phi (t) $, $ 0 \leq t < \tau $, preceding $ a $, but is not possible to the point $ a = \phi ( \tau ) $, then $ a $ is a singular point for analytic continuation of the element $ (U( \zeta ,\ R),\ f _ \zeta ) $ along the curve $ L $. In this instance it will also be singular for continuation along all curves of the equivalence class $ \{ L \} $ which pass through $ a $. The pair $ (a, \{ L \} ) $, consisting of the point $ a \in \overline{\mathbf C} $ and the equivalence class $ \{ L \} $ of curves $ L $ which pass through $ a $ for each of which $ a $ is singular, is called a singular point of the analytic function $ f(z) $ defined by the element $ (U( \zeta ,\ R),\ f _ \zeta ) $. Two singular points $ (a, \{ L \} ) $ and $ (b, \{ M \} ) $ are said to coincide if $ a = b $ and if the classes $ \{ L \} $ and $ \{ M \} $ coincide. The point $ a $ of the extended complex plane $ \overline{\mathbf C} $ is then called the projection, or $ z $- coordinate, of the singular point $ (a, \{ L \} ) $; the singular point $ (a, \{ L \} ) $ is also said to lie above the point $ a \in \mathbf C $. In general, several (even a countable set of) different singular and regular pairs $ (a, \{ L \} ) $ obtained through analytic continuation of one and the same element $ (U( \zeta ,\ R),\ f _ \zeta ) $ may lie above one and the same point $ a \in \overline{\mathbf C} $( cf. Branch point).

If the radius of convergence of the initial series (1) $ R < \infty $, then on the boundary circle $ \Gamma = \{ {z \in \overline{\mathbf C} } : {| z- \zeta | = R} \} $ of the disc of convergence $ U( \zeta ,\ R) $ there lies at least one singular point $ a $ of the element $ (U( \zeta ,\ R),\ f _ \zeta ) $, i.e. there is a singular point of the analytic function $ f(z) $ for continuation along the curves $ z = \phi (t) $, $ 0 \leq t \leq 1 $, of the class $ \{ L \} $ such that $ z = \phi (t) \in U( \zeta ,\ R) $ when $ 0 \leq t < 1 $, $ a = \phi (1) $. In other words, a singular point of the element $ (U( \zeta ,\ R),\ f _ \zeta ) $ is a point $ a \in \Gamma $ such that direct analytic continuation of the element $ (U( \zeta ,\ R),\ f _ \zeta ) $ from the disc $ U( \zeta ,\ R) $ to any neighbourhood $ V(a) $ is impossible. In this situation, and generally in all cases where the lack of an obvious description of the class of curves $ \{ L \} $ cannot give rise to ambiguity, one usually restricts to the $ z $- coordinate $ a $ of the singular point. The study of the position of singular points of an analytic function, in dependence on the properties of the sequence of coefficients $ \{ c _{k} \} _{k=0} ^ \infty $ of the initial element $ (U( \zeta ,\ R),\ f _ \zeta ) $, is one of the main directions of research in function theory (see Hadamard theorem on multiplication; Star of a function element, as well as [1], [3], [5]). It is well-known, for example, that the singular points of the series $$ f _{0} (z) = \sum _{k=0} ^ \infty b ^{k} z ^ {d ^ k} , $$ where $ b \in \overline{\mathbf C} $, $ | b | < 1 $, and $ d \geq 2 $ is a natural number, fill the whole boundary $ \Gamma = \{ {z \in \overline{\mathbf C} } : {| z | = 1} \} $ of its disc of convergence $ U(0,\ 1) $, although the sum of this series is continuous everywhere in the closed disc $ \overline{U} (0,\ 1) = \{ {z \in \overline{\mathbf C} } : {| z | \leq 1} \} $. Here, $ \Gamma $ is the natural boundary of the analytic function $ f _{0} (z) $; analytic continuation of $ f _{0} (z) $ across the boundary of the disc $ U(0,\ 1) $ is impossible.

Suppose that in a sufficiently small neighbourhood $ V(a) = \{ {z \in \overline{\mathbf C} } : {| z-a | < R} \} $ of a point $ a \neq \infty $( or $ V( \infty ) = \{ {z \in \overline{\mathbf C} } : {| z | > R} \} $), analytic continuation along the curves of a specific class $ \{ L \} $ is possible to all points other than $ a $, for all elements obtained, i.e. along all curves situated in the deleted neighbourhood $ V ^ \prime (a) = \{ {z \in \overline{\mathbf C} } : {0< | z-a | < R} \} $( respectively, $ V ^ \prime ( \infty ) = \{ {z \in \mathbf C} : {| z-a | > R} \} $); the singular point $ (a, \{ L \} ) $ is then called an isolated singular point. If analytic continuation of the elements obtained along the curves of the class $ \{ L \} $ along all possible closed curves situated in $ V ^ \prime (a) $ does not alter these elements, then the isolated singular point $ (a, \{ L \} ) $ is called a single-valued singular point. This type of singular point can be a pole or an essential singular point: If an infinite limit $ \mathop{\rm lim}\nolimits \ f(z) = \infty $ exists when $ z $ moves towards $ a $ along the curves of the class $ \{ L \} $, then the single-valued singular point $ (a, \{ L \} ) $ is called a pole (of a function); if no finite or infinite limit $ \mathop{\rm lim}\nolimits \ f(z) $ exists when $ z $ moves towards $ a $ along the curves of the class $ \{ L \} $, then $ (a, \{ L \} ) $ is called an essential singular point; the case of a finite limit corresponds to a regular point $ (a, \{ L \} ) $. If analytic continuation of the elements obtained along the curves of the class $ \{ L \} $ along closed curves surrounding $ a $ in $ V ^ \prime (a) $ alters these elements, then the isolated singular point $ (a, \{ L \} ) $ is called a branch point or a many-valued singular point. The class of branch points is in turn subdivided into algebraic branch points and transcendental branch points (including logarithmic branch points, cf. Algebraic branch point; Logarithmic branch point; Transcendental branch point). If after a finite number $ m \geq 2 $ of single loops around $ a $ in the same direction within $ V ^ \prime (a) $, the elements obtained along the curves of the class $ \{ L \} $ take their original form, then $ (a, \{ L \} ) $ is an algebraic branch point and the number $ m-1 $ is called its order. Conversely, when the loops around $ a $ give more and more new elements, $ (a, \{ L \} ) $ is a transcendental branch point.

For example, for the function $$ f(z) = \frac{1}{(1+z ^{1/2} )(1 + z ^{1/6} )} $$ the points $ a = 0 $, $ \infty $( for all curves) are algebraic branch points of order 5. As a point function, $ f(z) $ can be represented as a single-valued function only on the corresponding Riemann surface $ S $, consisting of 6 sheets over $ \overline{\mathbf C} $ joined in a specific way above the points $ 0,\ \infty $. Moreover, three proper branches of $ f(z) $ lie above the point $ a=1 $, which are single-valued on the three corresponding sheets of $ S $; on one sheet of $ S $ there is a pole of the second order, and on two sheets of $ S $ there are poles of the first order. In general, the introduction of the concept of a Riemann surface is particularly convenient and fruitful when studying the character of a singular point.

If the radius of convergence of the initial series (1) $ R = \infty $, then it represents an entire function $ f(z) $, i.e. a function holomorphic in the entire finite plane $ \mathbf C $. When $ f(z) \neq \textrm const $, this function has a single isolated singular point $ a = \infty $ of single-valued character; if $ a = \infty $ is a pole, then $ f(z) $ is an entire rational function, or a polynomial; if $ a = \infty $ is an essential singular point, then $ f(z) $ is a transcendental entire function.

A meromorphic function $ f(z) $ in the finite plane $ \mathbf C $ is obtained when analytic continuation of the series (1) leads to a single-valued analytic function $ f(z) $ in $ \mathbf C $ all singular points of which are poles. If $ a = \infty $ is a pole or a regular point, then the total number of poles of $ f(z) $ in the extended plane $ \overline{\mathbf C} $ is finite and $ f(z) $ is a rational function. For a transcendental meromorphic function $ f(z) $ in $ \mathbf C $, the point at infinity $ a = \infty $ can be a limit point of the poles — this is the simplest example of a non-isolated singular point of a single-valued analytic function. A meromorphic function in an arbitrary domain $ D \subset \overline{\mathbf C} $ is defined in the same way.

Generally speaking, the projections of non-isolated singular points can form different sets of points in the extended complex plane $ \overline{\mathbf C} $. In particular, whatever the domain $ D \subset \overline{\mathbf C} $, an analytic function $ f _{D} (z) $ exists in $ D $ for which $ D $ is its natural domain of existence, and the boundary $ \Gamma = \partial D $ is its natural boundary; thus, analytic continuation of the function $ f _{D} (z) $ across the boundary of $ D $ is impossible. Here, the natural boundary $ \Gamma $ consists of accessible and inaccessible points (see Limit elements). If a point $ a \in \Gamma $ is accessible along the curves of a class $ \{ L \} $( there may be several of these classes), all situated in $ D $ except for the end point $ a $, then only singular points of the function $ f _{D} (z) $ can lie above $ a $, since if this were not the case, analytic continuation of $ f _{D} (z) $ across the boundary of $ D $ through a part of $ \Gamma $ in a neighbourhood of $ a $ would be possible; the accessible points form a dense set on $ \Gamma $.


The role of the defining element of an analytic function $ f(z) $ of several complex variables $ z = (z _{1} \dots z _{n} ) $, $ n > 1 $, can be played by, for example, a Weierstrass element $ (U ^{n} ( \zeta ,\ R),\ f _ \zeta ) $ in the form of a multiple power series $$ \tag{2} f _ \zeta = f _ \zeta (z) = $$ $$ = \sum _ {k _{1} \dots k _{n} =0} ^ \infty c _ {k _{1} \dots k _ n} (z _{1} - \zeta _{1} ) ^ {k _ 1} \dots (z _{n} - \zeta _{n} ) ^ {k _ n} $$ and the polydisc of convergence of this series $$ U ^{n} ( \zeta ,\ r) = \{ {z \in \mathbf C ^ n} : {| z _{v} - \zeta _{v} | < R _{v} , v = 1 \dots n} \} , $$ with centre $ \zeta = ( \zeta _{1} \dots \zeta _{n} ) \in \mathbf C ^{n} $ and radius of convergence $ R = \{ R _{1} > 0 \dots R _{n} > 0 \} $. By taking in the process of analytic continuation of the element (2) along all possible curves $ L: \ [0,\ 1] \rightarrow \mathbf C ^{n} $, mappings of the interval $ 0 \leq t \leq 1 $ into the complex space $ \mathbf C ^{n} $ as basis, a general definition of the singular points $ (a, \{ L \} ) $, $ a \in \mathbf C ^{n} $, of the function $ f(z) $ is obtained, which is formally completely analogous to the one mentioned above for the case $ n=1 $.


However, as a result of the overdeterminacy of the Cauchy-Riemann equations when $ n > 1 $ and the resulting "large power" of analytic continuation, the case $ n > 1 $ differs radically from the case $ n=1 $. In particular, for $ n > 1 $ there are domains $ D \subset \mathbf C ^{n} $ which cannot be natural domains of existence of any single-valued analytic or holomorphic function. In other words, on specific sections of the boundary $ \partial D $ of this domain there are no singular points of any holomorphic function $ f(z) $ defined in $ D $, and analytic continuation is possible across them. For example, the Osgood–Brown theorem holds: If a compact set $ K $ is situated in a bounded domain $ D \subset \mathbf C ^{n} $ such that $ D \setminus K $ is also a domain, and if a function $ f(z) $ is holomorphic in $ D \setminus K $, then it can be holomorphically continued onto the whole domain $ D $( see also Removable set). The natural domains of existence of holomorphic functions are sometimes called domains of holomorphy (cf. Domain of holomorphy), and are characterized by specific geometric properties. Analytic continuation of a holomorphic function $ f(z) $ which is originally defined in a domain $ D \subset \mathbf C ^{n} $ while retaining its single-valuedness makes it necessary to introduce, generally speaking, many-sheeted domains of holomorphy over $ \mathbf C ^{n} $, or Riemann domains — analogues of Riemann surfaces (cf. Riemannian domain). In this interpretation, the singular points of a holomorphic function $ f(z) $ prove to be points of the boundary $ \Gamma = \partial \widehat{D} $ of its domain of holomorphy $ \widehat{D} $. The Osgood–Brown theorem shows that the connected components of $ \Gamma $ cannot form compact sets $ K $ such that the function $ f(z) $ is holomorphic in $ \widehat{D} \setminus K $. In particular, for $ n > 1 $ there do not exist isolated singular points of holomorphic functions.

The simplest types of singular points of analytic functions of several complex variables are provided by meromorphic functions $ f(z) $ in a domain $ D \subset \mathbf C ^{n} $, $ n \geq 1 $, which are characterized by the following properties: 1) $ f(z) $ is holomorphic everywhere in $ D $ with the exception of a polar set $ P $, which consists of singular points; and 2) for any point $ a \in P $ there are a neighbourhood $ V(a) $ and a holomorphic function $ \psi _{a} (z) $ in $ V(a) $ such that the function $ \phi _{a} (z) = \psi _{a} (z)f(z) $ can be continued holomorphically to $ V(a) $. The singular points $ a \in P $ are then divided into poles, at which $ \phi _{a} (a) \neq 0 $, and points of indeterminacy, at which $ \phi _{a} (a) = 0 $. In the case of a pole, $ \mathop{\rm lim}\nolimits \ f(z) = \infty $ when $ z $ moves towards $ a $, $ z \in D \setminus P $; in any neighbourhood of a point of indeterminacy, $ f(z) $ takes all values $ w \in \mathbf C $. For example, the meromorphic function $ f(z) = z _{1} /z _{2} $ in $ \mathbf C ^{2} $ has the straight line $ P = \{ {z = (z _{1} ,\ z _{2} ) \in \mathbf C ^ 2} : {z _{2} = 0} \} $ as its polar set; all points of this straight line are poles, with the exception of the point of indeterminacy $ (0,\ 0) $. A meromorphic function $ f(z) $ in its domain of holomorphy $ \widehat{D} $ can be represented globally in $ \widehat{D} $ as the quotient of two holomorphic functions, i.e. its polar set $ P $ is an analytic set.

A point $ a \in \mathbf C ^{n} $ is called a point of meromorphy of a function $ f(z) $ if $ f(z) $ is meromorphic in a certain neighbourhood of that point; thus, if a singular point is a point of meromorphy, then it is either a pole or a point of indeterminacy. All singular points of the analytic function $ f(z) $ which are not points of meromorphy are sometimes called essential singular points. These include, for example, the branch points of $ f(z) $, i.e. the branching points of its (many-sheeted) domain of holomorphy $ \widehat{D} $. The dimension of the set of all singular points of a holomorphic function $ f(z) $ is, in general, equal to $ 2n-1 $. Given certain extra restrictions on $ f(z) $ this set proves to be analytic (and, consequently, is of smaller dimension; see [2]).

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 1–2 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001
[3] S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian)
[4] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) MR0173749 Zbl 0135.12101
[5] L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3 MR0068621 Zbl 0064.06902
[6] L. Bieberbach, "Lehrbuch der Funktionentheorie" , 1–2 , Chelsea, reprint (1945) MR0012116 Zbl 0060.19907
[7] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[8] B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian) MR0155003
[9] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) MR0180696 Zbl 0141.08601
[10] H. Behnke, P. Thullen, "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer (1970) (Elraged & Revised Edition. Original: 1934)


Comments

Sometimes a point at which an analytic function $ f(z) $ is not defined but at which it can be defined so as to remain analytic is called a removable singular point.

If $ n =1 $, a famous theorem concerning values of an analytic function in a neighbourhood of an essential singular point is Picard's great theorem (cf. Picard theorem). See Fabry theorem for restrictions on the coefficients and powers so that $ f(z) $ has a singular point.

The Osgood–Brown theorem is also known as the Hartogs extension theorem.

References

[a1] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4 MR0344507 Zbl 0271.32001
[a2] H. Cohn, "Conformal mapping on Riemann surfaces" , Dover, reprint (1980) MR0594937 Zbl 0493.30001
[a3] J.B. Conway, "Functions of one complex variable" , Springer (1978) MR0503901 Zbl 0887.30003 Zbl 0277.30001
[a4] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 6 MR0635928 Zbl 0471.32008
[a5] R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) MR0847923
[a6] E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) MR1111477 Zbl 0683.32002
[a7] R. Narasimhan, "Several complex variables" , Univ. Chicago Press (1971) MR0342725 Zbl 0223.32001
[a8] R. Remmert, "Funktionentheorie" , 1 , Springer (1984) MR0753290 Zbl 0542.30001
[a9] B. Kaupp, "Holomorphic functions of several variables" , de Gruyter (1983) (Translated from German)

A singular point, or singularity, of an algebraic variety is a point at which smoothness is violated. More accurately, let $ X $ be an algebraic variety or a scheme of finite type over a field $ k $. Then a point $ x \in X $ is said to be singular if the corresponding local ring $ {\mathcal O} _{X,x} $ is not regular (regularity of a local Noetherian ring $ A $ with maximal ideal $ \mathfrak m $ means that $ \mathop{\rm dim}\nolimits \ \mathfrak m / \mathfrak m ^{2} = \mathop{\rm dim}\nolimits \ A $). The set of singular points of $ X $ is closed in the Zariski topology and is denoted by $ \mathop{\rm Sing}\nolimits \ X $. If $ X $ is a reduced variety, then $ \mathop{\rm Sing}\nolimits \ X $ is nowhere dense in $ X $. If $ x $ is an isolated point in $ \mathop{\rm Sing}\nolimits \ X $, then $ x $ is said to be an isolated singular point. In order to test whether or not $ x \in X $ is singular, Jacobi's criterion is used (see Smooth scheme).

A proper birational morphism $ \pi : \overline{X} \rightarrow X $, where $ \overline{X} $ is a smooth variety, is called a resolution of singularities (a desingularization) of the algebraic variety $ X $. The existence of a resolution of singularities has been proved for a broad class of varieties, in particular, for all varieties over a field of characteristic $ 0 $( see [13]). As a rule, it is not unique. A resolution of singularities is used to introduce various invariants of a variety $ X $; an example of this are the cohomology spaces $ H ^{i} ( \overline{X} ,\ {\mathcal O} _ {\overline{X} } ) $. A normal variety $ X $ for which $ H ^{i} ( \overline{X} ,\ {\mathcal O} _ {\overline{X} } ) = 0 $ for all $ i > 0 $ is called a variety with rational singularities. Toroidal singularities [6] and singularities of Schubert varieties [3] are rational. The dimension of the space $ H ^{n-1} ( \overline{X} ,\ {\mathcal O} _ {\overline{X} } ) $ for an $ n $- dimensional variety $ X $ is called the geometric genus of $ X $. See also Resolution of singularities.

The theory of deformations of singularities, i.e. of varieties with singular points, is constructed in parallel with the theory of deformations of (smooth) algebraic varieties. A flat morphism $ f: X \rightarrow S $ for which $ f ^ {\ -1} (s _{0} ) = X _{0} $ for a certain $ s _{0} \in S $ is called a deformation of $ X _{0} $; the space $ S $ is called the base of the deformation. For a variety $ X _{0} $ with an isolated singular point there is a versal deformation which contains all deformations of the variety $ X _{0} $. The singularity can be rigid, i.e. the base of the versal deformation consists of one point and all its deformations are trivial [4]. The opposite of rigid singular points are smoothable singular points, in the base $ S $ of the versal deformation of which are points such that $ X _{s} = f ^ {\ -1} (s) $ are non-singular. The set $ D $ of points $ s \in S $ with singular $ X _{s} $ is called the discriminant subset.

An important part in the study of deformations is played by the action of the monodromy group $ \pi _{1} (S \setminus D) $ on cohomology spaces of fibres of $ X $.


A proper morphism $ \pi : \ \overline{X} \rightarrow X $ is called a simultaneous resolution of singularities of a family $ X \rightarrow S $ if $ \overline{X} $ is a smooth $ S $- scheme, and for any $ s \in S $ the morphism $ \overline{X} _{s} \rightarrow X _{s} $ is a resolution of singularities. The versal deformation of simple singular points (see below) permits a simultaneous resolution after a certain finite covering of its base, whereby the Weyl group of the corresponding root system serves as the Galois group of the covering (see [5]).

Singular points of a complex hypersurface.

Let a hypersurface $ X $ be defined in $ \mathbf C ^{n+1} $ by one equation $ f(x _{0} \dots x _{n} ) = 0 $, where $ f $ is a polynomial (or germ of an analytic function at the point $ 0 $). The ideal $ J(f \ ) = ( \partial f/ \partial x _{0} \dots \partial f/ \partial x _{n} ) $ in the ring $ \mathbf C \{ x _{0} \dots x _{n} \} $ is called the Jacobi ideal of the polynomial $ f $; the singular point 0 is isolated if and only if the space $ \mathbf C \{ x _{0} \dots x _{n} \} /J(f \ ) $ has finite dimension. The dimension $ \mu $ of this space is called the Milnor number of the polynomial $ f $, and coincides with the rank of the free Abelian group $ H _{n} (X _ \epsilon ,\ \mathbf Z ) $, where $ X _ \epsilon $ is defined by the equation $ f(x _{0} \dots x _{n} ) = \epsilon $, for small $ \epsilon \neq 0 $. More precisely, the manifold $ X _ \epsilon $ is homotopically equivalent to a bouquet of $ \mu $ spheres of dimension $ n $( see [12]). The base of the versal deformation of this singularity is non-singular and is also of dimension $ \mu $( see [9]). The simplest example is a non-degenerate quadratic singularity $ x _{0} ^{2} + \dots + x _{n} ^{2} = 0 $; here $ \mu = 1 $.


A singularity such that in a deformation only a finite number of other singularities appear is called a simple singular point of the hypersurface [9]; the hypersurface is then defined by one of the following equations:

$ A _ \mu : $: $ x _{0} ^ {\mu + 1} + x _{1} ^{2} + \dots + x _{n} ^{2} = 0 $, $ \mu \geq 1 $;


$ D _ \mu $: $ x _{0} ^ {\mu -1} + x _{0} x _{1} ^{2} + x _{2} ^{2} + \dots + x _{n} ^{2} = 0 $, $ \mu \geq 4 $;


$ E _{6} $: $ x _{0} ^{4} + x _{1} ^{3} + x _{2} ^{2} + \dots + x _{n} ^{2} = 0 $;


$ E _{7} $: $ x _{0} ^{3} x _{1} + x _{1} ^{3} + x _{2} ^{2} + \dots + x _{n} ^{2} = 0 $;


$ E _{8} $: $ x _{0} ^{5} + x _{1} ^{3} + x _{2} ^{2} + \dots + x _{n} ^{2} = 0 $.


The lower index $ \mu $ is the Milnor number of the singularity. In the case of surfaces $ (n=2) $ these singularities are called Du Val singularities, or rational singular double points. These singular points can also be characterized by the fact that the intersection form on the space $ H _{n} (X _ \epsilon ,\ \mathbf R ) $ is definite. Next, in order of complexity, the unimodal singularities are classified [9]. Natural analogues of these concepts are studied, as well as their connection with catastrophe theory [10]. Many theorems on the singular points of hypersurfaces are extended to the singular points of complete intersections.

Singular points of curves.

Let $ A $ be the local ring of a singular point $ x $ of a curve, and let $ \overline{A} $ be its normalization; a principal invariant of a singular point is $ \delta _{x} = \mathop{\rm dim}\nolimits \ \overline{A} /A $. For an irreducible curve $ X $, its arithmetic genus is equal to the geometric genus plus $ \sum _{x} \delta _{x} $( summation over all singular points of $ X $). Hence, for a plane curve $ 2 \delta _{x} = \mu + r - 1 $, where $ \mu $ is the Milnor number while $ r $ is the number of branches of the curve at the point $ x $.


Let $ X \subset \mathbf C ^{2} $ be a plane irreducible curve with a singularity of multiplicity $ n $ at the point 0 (see Multiplicity of a singular point). $ X $ then permits a parametrization $ x = t ^{n} $, $ y = \sum _{i\geqn} a _{i} t ^{i} $, which is written in the form $$ y = \sum _{i} a _{i} x ^{i/n} $$( a Puiseux expansion). The numbers $$ \frac{m _ 1}{n _ 1} < \frac{m _ 2}{n _{1} n _ 2} < \dots < \frac{m _ g}{n _{1} \dots n _ g} = \frac{m _ g}{n} $$ are called the characteristic exponents of this expansion, where $ m _{1} /n _{1} $ is the first non-integer exponent in the Puiseux expansion, $ m _{2} /n _{1} n _{2} $ is the first exponent indivisible by $ 1/n _{1} $, etc. The sequence $ \{ n,\ \beta _{1} \dots \beta _{g} \} $, where $ \beta _{v} = (m _{v} n)/(n _{1} \dots n _{v} ) $, is called the characteristic of the singularity. Plane one-dimensional singularities are topologically equivalent if and only if their characteristics coincide (see [8]).

Singular points of surfaces.

Among the resolutions of singularities of normal surfaces, the minimal resolutions $ \pi : \ \overline{X} \rightarrow X $ through which all remaining resolutions pass, are uniquely distinguished. If $ x $ is a singular point of a surface $ X $, then the curve $ A = \pi ^{-1} (x) $ is said to be exceptional. The weight graph $ \Gamma $ of the curve $ A $ is a combinatorial invariant of the singular point $ x $, and its vertices correspond to the irreducible components $ A _{i} $ of $ A $; the intersection points of the components $ A _{i} $ and $ A _{j} $ are represented by the edges between the corresponding vertices; a weight equal to the genus of the curve $ A _{i} $, and sometimes even to the index of self-intersection $ (A _{i} ^{2} ) $, is attributed to the vertex. The matrix $ \| (A _{i} ,\ A _{j} ) \| $ of intersections of components of $ A $ is negative definite; the graph $ \Gamma $ is connected. The smallest positive divisor $ Z = \sum r _{i} A _{i} $ such that $ (Z,\ A _{i} ) \leq 0 $ for all $ i $ is called the fundamental cycle of the singularity. It always exists and its arithmetic genus $$ p(Z) = 1 - \mathop{\rm dim}\nolimits \ H ^{0} (Z,\ {\mathcal O} _{Z} ) + \mathop{\rm dim}\nolimits \ H ^{1} (Z,\ {\mathcal O} _{Z} ) $$ is non-negative. A singular point is rational if and only if $ p(Z) = 0 $; in this case its multiplicity equals $ -(Z ^{2} ) $, while the dimension of the tangent Zariski space is greater by one [1]. Elliptic singularities (i.e. singular points with $ p(Z) = 1 $) are also studied [7].

References

[1] M. Artin, "On isolated rational singularities of surfaces" Amer. J. Math. , 88 (1966) pp. 129–136 MR0199191 Zbl 0142.18602
[2] A. Grothendieck (ed.) et al. (ed.) , Groupes de monodromie en géométrie algébrique. SGA 7 , Lect. notes in math. , 288 , Springer (1972) MR0354656
[3] G. Kempf, "On the collapsing of homogeneous bundles" Invent. Math. , 37 (1976) pp. 229–239 MR0424841 Zbl 0338.14015
[4] M. Schlessinger, "Rigidity of quotient singularities" Invent. Math. , 14 (1971) pp. 17–26 MR0292830 Zbl 0232.14005
[5] H. Pinkham, "Resolution simultanee de points doubles rationnels" M. Demazure (ed.) et al. (ed.) , Sem. sur les Singularités des Surfaces , Lect. notes in math. , 777 , Springer (1980) pp. 179–203 Zbl 0457.14004
[6] G. Kempf (ed.) et al. (ed.) , Toroidal embeddings , Lect. notes in math. , 339 , Springer (1973) MR0335518 Zbl 0271.14017
[7] S.S-T. Yau, "On maximally elliptic singularities" Trans. Amer. Math. Soc. , 257 (1980) pp. 269–329 MR0552260 Zbl 0343.32009
[8] O. Zariski, "Studies in equisingularity III. Saturation of local rings and equisingularity" Amer. J. Math. , 90 (1968) pp. 961–1023 MR0237493 Zbl 0189.21405
[9] V.I. Arnol'd, "Critical points of smooth functions and their normal forms" Russian Math. Surveys , 30 : 5 (1975) pp. 1–75 Uspekhi Mat. Nauk , 30 : 5 (1975) pp. 3–65 Zbl 0343.58001
[10] M. Golubitskii, V. Guillemin, "Stable mappings and their singularities" , Springer (1973) MR0467801
[11] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , 1–2 , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[12] J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968) MR0239612 Zbl 0184.48405
[13] H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero I, II" Ann. of Math. , 79 (1964) pp. 109–326 MR0199184 Zbl 0122.38603

V.I. Danilov

Comments

Let the hypersurface $ X $ in $ \mathbf C ^{n+1} $ be defined by $ f(x _{0} \dots x _{n} ) = 0 $ and let $ 0 $ be an isolated singular point. The fibration $ f: \ \mathbf C ^{n+1} \rightarrow D( \epsilon ) $, where $ D( \epsilon ) $ is a small disc of radius $ \epsilon $ around zero, with fibre $ X _ \epsilon = \{ {(x _{0} \dots x _{n} )} : {f(x _{0} \dots x _{n} ) = \epsilon} \} $ is called the Milnor fibration.

The ring $ \mathbf C \{ x _{0} \dots x _{n} \} $ is the ring of convergent power series in $ x _{0} \dots x _{n} $.


The invariant $ \delta _{x} $ of a singular point $ x $ on a curve intuitively counts the number of double points concentrated at $ x $, [a9].

Let $ f $ be a polynomial with isolated critical point at $ 0 \in \mathbf C ^{n+1} $. A Morsification of $ f $ is a polynomial mapping $ F: \ \mathbf C ^{n+2} \rightarrow \mathbf C $ such that $ F(z,\ 0) = f(z) $( i.e. $ F $ is a one-dimensional deformation of $ f \ $) and each $ f _ \lambda (z) = F(z,\ \lambda ) $ has only non-degenerate critical points in a neighbourhood of $ 0 \in \mathbf C ^{n+1} $ for small enough $ \lambda $. These always exist. Some examples (for $ n =2 $) are as follows, [a6], Sect. 7. Let $$ \Phi _ \mu (x,\ \lambda ) = $$ $$ = \left \{ \begin{array}{ll} {(x + \lambda ) ^{2} \dots (x + k \lambda ) ^ 2} & \textrm{ if } \mu = 2k, \\ {(x+ \lambda ) ^{2} \dots (x+k \lambda ) ^{2} (x+(k+1) \lambda )} & \textrm{ if } \mu = 2k+1 , \\ \end{array} \right .$$ $ A _ \mu $: $ f(x,\ y) = x ^ {\mu +1} -y ^{2} $, $ F(x,\ y,\ \lambda ) = \Phi _ {\mu +1} (x,\ \lambda ) - y ^{2} $;


$ D _ \mu $: $ f(x,\ y) = x ^{m-1} -xy ^{2} = x(x ^{m-2} -y ^{2} ) $, $ F(x,\ y,\ \lambda ) = x \Phi _ {\mu -2} (x,\ \lambda ) -xy ^{2} $;


$ E _{6} $: $ f(x,\ y) = x ^{3} + y ^{4} $, $ F(x,\ y,\ \lambda ) = (x- \mu )(x ^{2} - \lambda y ^{2} ) + y ^{4} $;


$ E _{7} $: $ f(x,\ y) = x ^{3} + xy ^{3} $, $ F(x,\ y,\ \lambda ) = (x- \mu )(x ^{2} + y ^{3} + \lambda y ^{2} - 6 \lambda xy) $,


where in the last two examples $ \mu $ is chosen small enough as a function of $ \lambda $. For $ E _{8} $ cf. [a6].

It is of course not an accident that the simple hypersurface singularities received the Dynkin diagram labels $ A _ \mu $, $ D _ \mu $, $ E _{6} $, $ E _{7} $, $ E _{8} $. In the examples given above the separatrix diagrams of the Morsifications are in fact the corresponding Dynkin diagrams. (But this is not necessarily the case for every Morsification.) Here the separatrix diagram of $ F(x,\ y,\ \lambda ) $ consists of the critical points of $ F(x,\ y,\ \lambda ) $ for some fixed $ \lambda \neq 0 $( and a number of lines joining these points (the vertices), where there is a line joining two critical points if and only if there is an integral curve of the gradient vector field $$ \left ( \frac{\partial F( x,\ y ,\ \lambda )}{\partial x} ,\ \frac{\partial F (x,\ y,\ \lambda )}{\partial y} \right ) $$ joining these points).

Cf. [8], [a1][a7], [a10], [a11] for more material on the interrelations between Dynkin diagrams and singularities and on other occurrences of Dynkin diagrams (the ADE problem).

References

[a1] V.I. Arnol'd, "Critical points of smooth functions" R. James (ed.) , Proc. Internat. Congress Mathematicians (Vancouver, 1974) , 1 , Canad. Math. Congress (1975) pp. 19–39 Zbl 0343.58001
[a2] E. Brieskorn, "Singular elements of semisimple algebraic groups" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 2 , Gauthier-Villars (1971) pp. 279–284 MR437798
[a3] E. Brieskorn, "Singularitäten" Jahresber. Deutsch. Math. Verein. , 78 (1976) pp. 93–112 MR0430301 Zbl 0334.32011
[a4] S.M. Gusein-Zade, "Dynkin diagrams for singularities of functions of two variables" Funct. Anal. Appl. , 8 : 4 (1974) pp. 295–300 Funkts. Anal. i Prilozh. , 8 : 4 (1974) pp. 23–30
[a5] S.M. [S.M. Gusein-Zade] Husein-Zade, "The monodromy groups of isolated singularities of hypersurfaces" Russian Math. Surveys , 32 : 2 (1977) pp. 23–65 Uspekhi Mat. Nauk , 32 : 2 (1977)
[a6] M. Hazewinkel, W. Hesselink, D. Siersma, F.D. Veldkamp, "The ubiquity of Coxeter–Dynkin diagrams" Nieuw Archief voor Wiskunde , 25 (1977) pp. 257–307 MR0491673 Zbl 0377.20037
[a7] K. Gawedzki, "Conformal field theory" , Sem. Bourbaki 1988/89 , Exp. 704 , Soc. Math. France (1989) pp. 95–126 MR1056302 MR1040570 MR1030458 Zbl 0699.53086 Zbl 0699.53085
[a8] E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German) MR0886476 Zbl 0588.14019
[a9] J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) pp. 65 MR0103191
[a10] P.J. Slodowy, "Simple singularities and simple algebraic groups" , Lect. notes in math. , 815 , Springer (1980) MR0584445 Zbl 0441.14002
[a11] V.I. Arnol'd, "Singularities of caustics and wave fronts" , Kluwer (1990) Zbl 0734.53001

A singular point of a vector field $ X $ is a point $ a $ for which $ X(a) = 0 $. A singular point is isolated if $ X $ does not vanish at points other than $ a $ in a sufficiently small neighbourhood of $ a $. A singular point is non-degenerate if $$ \mathop{\rm det}\nolimits \ \left \| \frac{\partial x ^ i}{\partial a ^ j} \ \right \| \neq 0. $$ A non-degenerate singular point is always isolated.

M.I. Voitsekhovskii

Comments

References

[a1] V.I. Arnol'd, "Singularities of caustics and wave fronts" , Kluwer (1990) Zbl 0734.53001

A singular point of a differential equation $$ \tag{1} X(x,\ y) \ dy = Y(x,\ y) \ dx $$ is any point $ (x _{0} ,\ y _{0} ) \in G $ which satisfies the condition $$ \tag{2} X(x _{0} ,\ y _{0} ) = Y(x _{0} ,\ y _{0} ) = 0; $$ here $ X,\ Y: \ G \rightarrow \mathbf R $ are continuous functions in a certain domain $ G \subset \mathbf R ^{2} $. The points of $ G $ which do not satisfy condition (2) are called the ordinary points of the equation (1). The point $ (x _{0} ,\ y _{0} ) \in G $ is sometimes also called a singular point of the equation (1) if condition (2) is not fulfilled but if the Cauchy problem for equation (1) with initial data $ (x _{0} ,\ y _{0} ) $ has more than one solution.

Equation (1) is a particular case of a system of differential equations in symmetric form: $$ \tag{3} \frac{dx _ 1}{X _{1} (x)} = \dots = \frac{dx _ n}{X _{n} (x)} , $$ where $ n \geq 2 $, $ x = (x _{1} \dots x _{n} ) $ and the functions $ X _{i} : \ G \rightarrow \mathbf R $, $ i = 1 \dots n $, are continuous in a domain $ G \subset \mathbf R ^{n} $. A point $ x _{0} \in G $ is called a singular point of the system (3) if $ X _{i} (x _{0} ) = 0 $, $ i = 1 \dots n $. In the opposite case $ x _{0} $ is an ordinary point of this system.

Let $ H $ be the set of singular points of the system (3) in the domain $ G $. If $ x _{0} \in G \setminus H $, then an index $ i _{0} \in \{ 1 \dots n \} $ and a neighbourhood $ U $ of the point $ x _{0} $ exist such that the system (3) can be represented in $ U $ in the normal form $$ \frac{dx _ i}{dx _ {i _ 0}} = f _{i} (x) , f _{i} \in C (U), i \neq i _{0} . $$ Thus, the behaviour of the integral curves of the system (3) in a neighbourhood of an ordinary point is described by theorems of the general theory of ordinary differential equations. In particular, the following parallelizability theorem holds: If through every point $ x _{0} $ of the set $ G \setminus H $ passes a unique integral curve of the system (3), then every point of this set has a neighbourhood $ V $ such that the family of arcs of integral curves of the system (3) which fill $ V $ is homeomorphic (and if $ X _{i} \in C ^{1} (G) $, $ i = 1 \dots n $, diffeomorphic) to a family of parallel straight lines.

If $ x _{0} \in H $, then no pair $ (i _{0} ,\ U) $ exists which possesses the above property, and the integral curves of the system (3) can form different configurations around $ x _{0} $. Thus, for the equation $$ (ax + by) \ dy = (cx + ey) \ dx, $$ where $ a,\ b,\ c,\ e \in \mathbf R $, while the matrix $$ A = \left \| \begin{array}{ll} a & b \\ c & e \\ \end{array} \right \| $$ is non-degenerate, the position of integral curves in a neighbourhood of the point $ (0,\ 0) $ can be of the type of a saddle, a node, a centre, or a focus. The same name is then also given to the point $ (0,\ 0) $.


The system (3) can be seen as the result of the elimination of the time $ t $ from an autonomous system of differential equations $$ \tag{4} \dot{x} = X(x), x \in \mathbf R ^{n} , X = (X _{1} \dots X _{n} ). $$ If (4) is a system of class ( $ C $, uniqueness) in $ G $, i.e. $ X \in C (G) $, and a unique trajectory of the system passes through every point of the domain $ G $, then the points of the set $ H $ will be stationary points (cf. Equilibrium position) for this trajectory. These points are often called the singular points of this system, insofar as they are (by definition) singular points of the vector field $ X $. The integral curves of the system (3) situated in $ G \setminus H $ are trajectories of the system (4) other than the stationary positions.

Thus, the problem of the behaviour of integral curves of the system (3) in a neighbourhood of a singular point and the problem of the positioning of the trajectories of the system (4) in a neighbourhood of an equilibrium position are equivalent. Research into these problems follows two main directions.

The first course, which has its origins in the work of H. Poincaré , aims to explain the possible topological types of how the trajectories of the system (4) are situated in a neighbourhood of an isolated stationary point (which can always be considered to coincide with the origin of the coordinates $ O $ $ (x = 0) $) and to discover the analytic criteria needed to distinguish them. The most complete results have been obtained for the case where the system (4) can be represented in the form $$ \tag{5} \dot{x} = Ax + f(x) , $$ where $ A $ is a constant non-degenerate matrix and $ f(x) = o( \| x \| ) $ when $ \| x \| \rightarrow 0 $. In this case the point $ O $ is said to be a simple, or non-degenerate, singular point of the system (4). The following Grobman–Hartman theorem has been established for the system (5): If the matrix $ A $ does not have purely imaginary eigenvalues, while the function $ f \in C ^{1} (G) $, then there is a homeomorphism $ h $ of a neighbourhood $ U $ of the point $ O $ onto a neighbourhood $ V $ of the same point which transfers the trajectories of the system (5) to the trajectories of the linear system $$ \tag{6} \dot{x} = Ax. $$ The homeomorphism $ h: \ U \rightarrow V $ which realizes a topological correspondence between the trajectories of the systems (5) and (6) is not a diffeomorphism, in general (nor can it be replaced by one).

Under the conditions of this theorem, the stationary point $ O $ of the system (5) is of the same topological type as the stationary point $ O $ of the system (6). In particular, for a system of the second order, it will be a saddle if the eigenvalues $ \lambda _{1} ,\ \lambda _{2} $ of the matrix $ A $ satisfy the condition $ \lambda _{1} \lambda _{2} < 0 $, and a topological node (node or focus) if $ \lambda _{1} \lambda _{2} > 0 $( given purely imaginary $ \lambda _{1} ,\ \lambda _{2} $, the point $ O $ for the system (6) is a centre, while for the system (5) it is either a centre, a focus or a centre-focus, cf. Centro-focus; Centre and focus problem; Saddle node; Node; Focus).

If the matrix $ A $ has purely imaginary or zero eigenvalues, then there are, in general, no topological equivalences between the systems (5) and (6) in a neighbourhood of the point $ O $. Under these conditions, the behaviour of the trajectories of the system (5) in a neighbourhood of the point $ O $ has been studied in great detail in those cases where the matrix $ A $ has at most two eigenvalues with zero real parts while the function $ f $ is analytic. In particular, for a system of the second order with a non-zero matrix $ A $, all possible topological types of positioning of trajectories in a neighbourhood of $ O $ are clarified, and the coefficient criteria needed to distinguish between them have been given, up to the distinction between a centre and a focus [9]. Here, apart from a saddle, topological node or centre, the point $ O $ can be a saddle with two separatrices, a saddle-node (a neighbourhood $ U $ of the point $ O $ is divided by three trajectories (separatrices) adjoining $ O $ into three sectors: two hyperbolic sectors, filled by trajectories which leave $ U $ at both ends, and one parabolic sector, filled by trajectories which leave $ U $ at one end, while the other approaches $ O $) or a point with elliptical sector (a neighbourhood $ U $ of this point is divided into 4 sectors: one hyperbolic, two parabolic and one elliptic, filled by trajectories which approach $ O $ at both ends). For a system of the second order with a zero matrix $ A $, algorithms for the resolution of singularities have been worked out (see, for example, Frommer method or local methods in [12]) which, with the aid of a finite number of steps of the resolution process, give a clarification of the topological type of the point $ O $, accurate up to the solution of the problem of distinguishing between a centre and a focus. This problem (see Centre and focus problem) arises for a system of the second order in the form (5) when the matrix $ A $ has purely imaginary eigenvalues, and can arise in the case of two zero eigenvalues of this matrix. It is solved for particular classes of such systems (see, e.g., [14]).

An important characteristic of the isolated stationary point $ O $ of the system (4) is its Poincaré index. For $ n=2 $ it is defined as the rotation of the vector field $ X $ around the point $ O $( cf. Rotation of a vector field) along a circle $ \| x \| = \rho $ of a sufficiently small radius $ \rho $ in the positive direction, measured in units of a complete revolution. For example, the index of a simple saddle is equal to $ -1 $, the index of a node, focus or centre is equal to 1. When $ n $ is arbitrary, the index of the point $ O $ is defined as the degree of the mapping $ h $( cf. Degree of a mapping) of the sphere $ \| x \| = \rho $ of a sufficiently small radius $ \rho $ onto itself, defined by the formula: $$ h(x) = \frac{\rho X (x)}{\| X(x) \|} . $$ This course of research has led to the general qualitative theory of differential equations, while the emphasis of the research has shifted from local to global problems — the study of the behaviour of the trajectories of the system (4) in the entire domain $ G $, which is taken more and more often as a smooth manifold of some kind.

The other course of research, based on the work of A.M. Lyapunov [2], deals with studies of stability of solutions of systems of the form (4) (especially of equilibrium positions), as well as of non-autonomous systems of differential equations. This research is one of the branches of the theory of stability of motion (see Stability theory).

In complex analysis, the concept of a singular point is introduced for a differential equation $$ \tag{7} \frac{d ^{n} w}{dz ^ n} = P \left ( z,\ w,\ \frac{dw}{dz} \dots \frac{d ^{n-1} w}{dz ^ n-1} \right ) , $$ and also for a system of differential equations $$ \tag{8} \frac{dw}{dz} = P(z,\ w) , $$ where $ z $ is a complex variable, $ P $ is a rational function in $ w,\ w ^ \prime \dots w ^{(n-1)} $ or in the components $ w _{1} \dots w _{n} $ of the vector $ w $, $ n \geq 1 $, the coefficients of which are known analytic functions of $ z $. Any point $ z _{0} $ of the complex plane which is a singular point of at least one of the coefficients of the function $ P $ is said to be a singularity for equation (7) (for the system (8)) (see Singular point of an analytic function). Singular points of an equation or system, as a rule, are also singular for their solutions as analytic functions in $ z $. They are called fixed singular points (cf. Fixed singular point) of these solutions. Moreover, the solutions of equation (7) (system (8)) can have movable singular points (cf. Movable singular point), the position of which is determined by the initial data of the solution. Studies on various classes of equations of the form (7), (8), aimed at clarifying the analytic nature of the solutions in a neighbourhood of the singular points of the equations, and at clarifying the presence of movable singular points of various types in the solutions of these equations, is the subject of the analytic theory of differential equations.

References

[1a] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 7 (1881) pp. 375–422 Zbl 13.0591.01
[1b] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 8 (1882) pp. 251–296 Zbl 14.0666.01
[1c] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 1 (1885) pp. 167–244 Zbl 14.0666.01 Zbl 13.0591.01
[1d] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 2 (1886) pp. 151–217 Zbl 14.0666.01 Zbl 13.0591.01
[2] A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian) MR2559078 MR1229075 MR1179692 MR1154209 MR0569069 MR0039884 Zbl 0161.06303
[3] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) MR0121520 Zbl 0089.29502
[4] E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17 MR0069338 Zbl 0064.33002
[5] S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957) MR0094488 Zbl 0080.06401
[6] G. Sansone, R. Conti, "Non-linear differential equations" , Pergamon (1964) (Translated from Italian) MR0177153 Zbl 0128.08403
[7] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) MR0658490 Zbl 0476.34002
[8] V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian) Zbl 1049.34001 Zbl 0744.34001 Zbl 0659.58012 Zbl 0602.58020 Zbl 0577.34001 Zbl 0956.34502 Zbl 0956.34501 Zbl 0956.34503 Zbl 0237.34008 Zbl 0135.42601
[9] N.N. Bautin, E.A. Leontovich, "Methods and means for a qualitative investigation of dynamical systems on the plane" , Moscow (1976) (In Russian)
[10] V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) MR0100119
[11] N.P. Erugin, "A reader for a general course in differential equations" , Minsk (1979) (In Russian) MR0553638 MR0545337 Zbl 0446.34001
[12] A.D. [A.D. Bryuno] Bruno, "Local methods in nonlinear differential equations" , Springer (1989) (Translated from Russian) MR0993771 Zbl 0674.34002
[13] A.F. Andreev, "Singular points of differential equations" , Minsk (1979) (In Russian) MR0545898
[14] V.V. Amel'kin, N.A. Lukashevich, A.P. Sadovskii, "Non-linear oscillations in second-order systems" , Minsk (1982) (In Russian)

A.F. Andreev

Comments

References

[a1] M.A. Krasnosel'skii, A.I. [A.I. Perov] Perow, A.I. [A.I. Povoloskii] Powolzki, P.P. [P.P. Zabreiko] Sabrejko, "Vektorfelder in der Ebene" , Akademie Verlag (1966) (Translated from Russian) MR202121

A singular point of a differentiable mapping $ f $ is a point which is simultaneously irregular (critical) and improper for $ f $. More precisely, let $ M ^{m} $ and $ N ^{n} $ be two differentiable manifolds of dimensions $ m $ and $ n $, respectively, let $ f: \ M ^{m} \rightarrow N ^{n} $ be a differentiable mapping of the first onto the second, and let $ x ^{j} $ and $ y ^{i} = f(x ^{j} ) $ be local coordinates in them. If the rank of the matrix $ \| \partial y ^{i} / \partial x ^{j} \| $ at a point $ a \in M ^{m} $ is equal to $ m $, then the mapping $ f $ is said to be regular at $ a $. If the rank of the matrix $ \| \partial y ^{i} / \partial x ^{j} \| $ is equal to $ n $ at a point $ a \in M ^{m} $, then the mapping $ f $ is said to be proper at $ a $. At a singular point of $ f $, the rank of this matrix is not equal to $ m $ or $ n $. See also Singularities of differentiable mappings.

M.I. Voitsekhovskii

A singular point of a real curve $ F(x,\ y) = 0 $ is a point $ (x _{0} ,\ y _{0} ) $ at which the first partial derivatives vanish: $ (F _{x} ^ {\ \prime} ) _{0} = 0 $, $ (F _{y} ^ {\ \prime} ) _{0} = 0 $. A singular point is called a double point if at least one of the second partial derivatives of the function $ F(x,\ y) $ does not vanish. In studies on the structure of a curve in a neighbourhood of a singular point, the sign of the expression $$ \Delta = (F _{xx} ^ {\ \prime\prime} ) _{0} (F _{yy} ^ {\ \prime\prime} ) _{0} - (F _{xy} ^ {\ \prime\prime} ) _{0} ^{2} $$ is studied. If $ \Delta > 0 $, then the singular point is an isolated point (Fig.a); if $ \Delta < 0 $, it is a node (or point of self-intersection) (Fig.b); if $ \Delta = 0 $, then it is either an isolated point or is characterized by the fact that different branches of the curve have a common tangent at this point. If the branches of the curve are situated on different sides of the common tangent and on the same side of the common normal, then the singular point is called a cusp of the first kind (Fig.c); if the branches of the curve are situated on the same side of the common tangent and on the same side of the common normal, then the singular point is called a cusp of the second kind (Fig.d); if the branches are situated on different sides of the common normal and on different sides of the common tangent (Fig.e), or on the same side of the common tangent and on different sides of the common normal (Fig.f), then the singular point is called a point of osculation. See also Double point.

Figure: s085590a

Figure: s085590b

Figure: s085590c

Figure: s085590d

Figure: s085590e

Figure: s085590f

If all partial derivatives of the function $ F(x,\ y) $ up to order $ (k-1) $ inclusive vanish at a certain point and at least one of the derivatives of order $ k $ differs from zero, then this point is called a singular point of order $ k $( a multiple point).

Points which differ in any of their properties from other points of the curve are sometimes called singular points; see, for example, Point of inflection; Point of cessation; Breaking point; Point of rectification; Flat point.

A singular point of a spatial curve defined by the equations $ F(x,\ y,\ z) = 0 $, $ G(x,\ y,\ z) = 0 $ is a point in a neighbourhood of which the rank of the matrix $$ \left \| \begin{array}{lll} F _{x} ^ {\ \prime} &F _{y} ^ {\ \prime} &F _{z} ^ {\ \prime} \\ G _{x} ^ \prime &G _{y} ^ \prime &G _{z} ^ \prime \\ \end{array} \right \| $$ is less than two.

References

[1] P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian)
[2] S.S. Byushgens, "Differential geometry" , 1 , Moscow-Leningrad (1940) (In Russian)
[3] G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964) MR1191905 MR1056870 MR1056869 MR0887101 MR0845556 MR0845555 MR0524565 MR0473117 MR0344040 MR0344039 MR0238635 MR0238637 MR0238636 Zbl 0143.27002

A.B. Ivanov

A singular point of a real surface is a point of the surface $ x = x(u,\ v) $, $ y= y(u,\ v) $, $ z = z(u,\ v) $ at which the rank of the matrix $$ \left \| \begin{array}{lll} x _{u} &y _{u} &z _{u} \\ x _{v} &y _{v} &z _{v} \\ \end{array} \right \| $$ is less than two. If the surface is defined as the set of points whose coordinates satisfy an equation $ F(x,\ y,\ z) = 0 $, then a point $ (x _{0} ,\ y _{0} ,\ z _{0} ) $ of the surface at which the first partial derivatives of the function $ F(x,\ y,\ z) $ vanish is called a singular point: $$ (F _{x} ^ {\ \prime} ) _{0} = 0, (F _{y} ^ {\ \prime} ) _{0} = 0, (F _{z} ^ {\ \prime} ) _{0} = 0. $$ If not all second partial derivatives of the function $ F(x,\ y,\ z) $ vanish at the singular point, then the tangents of the surface at the singular point form a cone. If the tangent cone is non-degenerate, then the singular point is called a conic point; if the cone degenerates to two real planes, then the singular point is called a point of self-intersection of the surface; if the cone is imaginary, then the singular point is an isolated point of the surface.

Singular points can form so-called singular curves of a surface: an edge of regression, lines of self-intersection, lines of osculation, and others.

References

[1] A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian) MR0114163 Zbl 0084.17702
[2] A.P. Norden, "A short course of differential geometry" , Moscow (1958) (In Russian)
[3] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)

A.B. Ivanov

Comments

References

[a1] R.L. Bishop, S.I. Goldberg, "Tensor analysis on manifolds" , Dover, reprint (1980) MR0615912 Zbl 0218.53021
[a2] A. Pollack, "Differential topology" , Prentice-Hall (1974) MR0348781 Zbl 0361.57001
[a3] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5 MR0532834 MR0532833 MR0532832 MR0532831 MR0532830 MR0394453 MR0394452 MR0372756 MR1537051 MR0271845 MR0267467 Zbl 1213.53001 Zbl 0439.53005 Zbl 0439.53004 Zbl 0439.53003 Zbl 0439.53002 Zbl 0439.53001 Zbl 0306.53003 Zbl 0306.53002 Zbl 0306.53001 Zbl 0202.52201 Zbl 0202.52001
How to Cite This Entry:
Singular point. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Singular_point&oldid=44318
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article