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For an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c0270901.png" />, a critical point of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c0270903.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c0270904.png" /> of the complex plane at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c0270905.png" /> is regular but its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c0270906.png" /> has a zero of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c0270907.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c0270908.png" /> is a natural number. In other words, a critical point is defined by the conditions
+
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<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/c/c027/c027090/c0270909.png" /></td> </tr></table>
+
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 +
{{TEX|done}}
  
A critical point at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709010.png" />, of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709011.png" />, for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709012.png" /> which is regular at infinity, is defined by the conditions
+
For an analytic function  $  f ( z) $,
 +
a critical point of order  $  m $
 +
is a point  $  a $
 +
of the complex plane at which  $  f ( z) $
 +
is regular but its derivative  $  f ^ { \prime } ( z) $
 +
has a zero of order $  m $,  
 +
where  $  m $
 +
is a natural number. In other words, a critical point is defined by the conditions
  
<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/c/c027/c027090/c02709013.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {z \rightarrow a }
 +
 +
\frac{f ( z) - f ( a) }{( z - a)  ^ {m} }
 +
  = 0,\ \
 +
\lim\limits _ {z \rightarrow a }
 +
 +
\frac{f ( z) - f ( a) }{( z - a)  ^ {m+} 1 }
 +
  \neq  0.
 +
$$
  
Under the analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709014.png" />, the angle between two curves emanating from a critical point of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709015.png" /> is increased by a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709017.png" /> is regarded as the complex potential of some planar flow of an incompressible liquid, a critical point is characterized by the property that through it pass not one but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709018.png" /> stream lines, and the velocity of the flow at a critical point vanishes. In terms of the inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709019.png" /> (i.e. the function for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709020.png" />), a critical point is an algebraic branch point of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709021.png" />.
+
A critical point at infinity, a = \infty $,  
 +
of order  $  m $,
 +
for a function $  f ( z) $
 +
which is regular at infinity, is defined by the conditions
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709022.png" /> of a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709023.png" />-dimensional irreducible analytic set
+
$$
 +
\lim\limits _ {z \rightarrow \infty }
 +
[ f ( z) - f ( \infty )]
 +
z  ^ {m}  = 0,\ \
 +
\lim\limits _ {z \rightarrow \infty }
 +
[ f ( z) - f ( \infty )]
 +
z ^ {m + 1 }  \neq  0.
 +
$$
  
<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/c/c027/c027090/c02709024.png" /></td> </tr></table>
+
Under the analytic mapping  $  w = f ( z) $,
 +
the angle between two curves emanating from a critical point of order  $  m $
 +
is increased by a factor  $  m + 1 $.
 +
If  $  f ( z) $
 +
is regarded as the complex potential of some planar flow of an incompressible liquid, a critical point is characterized by the property that through it pass not one but  $  m + 1 $
 +
stream lines, and the velocity of the flow at a critical point vanishes. In terms of the inverse function  $  z = \psi ( w) $(
 +
i.e. the function for which  $  f [ \psi ( w)] \equiv w $),
 +
a critical point is an algebraic branch point of order  $  m + 1 $.
  
the latter being defined in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709026.png" /> in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709027.png" /> by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709029.png" /> are holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709031.png" /> complex variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709032.png" />, is called a critical point if the rank of the Jacobian matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709035.png" />, is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709036.png" />. The other points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709037.png" /> are called regular. There are relatively few critical points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709038.png" />: They form an analytic set of complex dimension at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709039.png" />. In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709040.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709041.png" />, and the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709042.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709043.png" />, the dimension of the set of critical points is at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709044.png" />.
+
A point  $  a $
 +
of a complex  $  ( n - m) $-
 +
dimensional irreducible analytic set
 +
 
 +
$$
 +
M  = \
 +
\{ {z \in V } : {
 +
f _ {1} ( z) = \dots =
 +
f _ {m} ( z) = 0 } \}
 +
,
 +
$$
 +
 
 +
the latter being defined in a neighbourhood $  V $
 +
of $  a $
 +
in the complex space $  \mathbf C  ^ {n} $
 +
by the conditions $  f _ {1} ( z) = \dots = f _ {m} ( z) = 0 $,  
 +
where $  f _ {1} \dots f _ {m} $
 +
are holomorphic functions on $  V $
 +
in $  n $
 +
complex variables, $  z = ( z _ {1} \dots z _ {n} ) $,  
 +
is called a critical point if the rank of the Jacobian matrix $  \| \partial  f _ {j} / \partial  z _ {k} \| $,  
 +
$  j = 1 \dots m $,  
 +
$  k = 1 \dots n $,  
 +
is less than $  m $.  
 +
The other points of $  M $
 +
are called regular. There are relatively few critical points on $  M $:  
 +
They form an analytic set of complex dimension at most $  n - m - 1 $.  
 +
In particular, when $  m = 1 $,  
 +
i.e. if $  M = \{ f _ {1} ( z) = 0 \} $,  
 +
and the dimension of $  M $
 +
is $  n - 1 $,  
 +
the dimension of the set of critical points is at most $  n - 2 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer (1964) {{MR|0173749}} {{ZBL|0135.12101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer (1964) {{MR|0173749}} {{ZBL|0135.12101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A point as described under 2) is also called a singular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709045.png" />, cf. [[#References|[a1]]].
+
A point as described under 2) is also called a singular point of $  M $,  
 +
cf. [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) pp. 95 (Translated from German) {{MR|0414912}} {{ZBL|0381.32001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) pp. 95 (Translated from German) {{MR|0414912}} {{ZBL|0381.32001}} </TD></TR></table>
  
A critical point of a smooth (i.e. continuously differentiable) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709046.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709047.png" />-dimensional differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709048.png" /> into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709049.png" />-dimensional differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709050.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709051.png" /> such that the rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709052.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709053.png" /> at this point (i.e. the dimension of the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709054.png" /> of the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709055.png" /> under the differential mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709056.png" />) is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709057.png" />. The set of all critical points is called the critical set, the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709058.png" /> of a critical point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709059.png" /> is called a critical value, and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709060.png" /> which is not the image of any critical point is called a regular point or a regular value (though it need not belong to the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709061.png" />); non-critical points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709062.png" /> are also called regular.
+
A critical point of a smooth (i.e. continuously differentiable) mapping $  f $
 +
of a $  k $-
 +
dimensional differentiable manifold $  M $
 +
into an $  l $-
 +
dimensional differentiable manifold $  N $
 +
is a point $  x _ {0} \in M $
 +
such that the rank $  \mathop{\rm Rk} _ {x _ {0}  }  f $
 +
of $  f $
 +
at this point (i.e. the dimension of the image $  df ( T _ {x _ {0}  } M) $
 +
of the tangent space to $  M $
 +
under the differential mapping $  df:  T _ {x _ {0}  } M \rightarrow T _ {f ( x _ {0}  ) } N $)  
 +
is less than $  l $.  
 +
The set of all critical points is called the critical set, the image $  f( x _ {0} ) $
 +
of a critical point $  x _ {0} $
 +
is called a critical value, and a point $  y \in N $
 +
which is not the image of any critical point is called a regular point or a regular value (though it need not belong to the image $  f( M) $);  
 +
non-critical points of $  M $
 +
are also called regular.
  
According to Sard's theorem, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709063.png" /> is smooth of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709065.png" />, then the image of the critical set is of the first category in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709066.png" /> (i.e. it is the union of at most countably many nowhere-dense sets) and has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709067.png" />-dimensional measure zero (see [[#References|[1]]], [[#References|[2]]]). The lower bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709068.png" /> cannot be weakened (see [[#References|[3]]]). The case most frequently needed is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709069.png" /> (in which case the proof is simplified, see [[#References|[4]]]). This theorem is widely used for reductions to [[General position|general position]] via "small movements" ; for example, it may readily be used to prove that, given two smooth submanifolds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709070.png" />, there exists an arbitrarily small translation of one of them such that their intersection will also be a submanifold (see [[#References|[2]]], [[#References|[4]]], and also [[Transversality|Transversality]] of mappings).
+
According to Sard's theorem, if $  f $
 +
is smooth of class $  C  ^ {m} $,
 +
$  m > \min ( k - l, 0) $,  
 +
then the image of the critical set is of the first category in $  N $(
 +
i.e. it is the union of at most countably many nowhere-dense sets) and has $  l $-
 +
dimensional measure zero (see [[#References|[1]]], [[#References|[2]]]). The lower bound for $  m $
 +
cannot be weakened (see [[#References|[3]]]). The case most frequently needed is $  m = \infty $(
 +
in which case the proof is simplified, see [[#References|[4]]]). This theorem is widely used for reductions to [[General position|general position]] via "small movements" ; for example, it may readily be used to prove that, given two smooth submanifolds in $  \mathbf R  ^ {n} $,  
 +
there exists an arbitrarily small translation of one of them such that their intersection will also be a submanifold (see [[#References|[2]]], [[#References|[4]]], and also [[Transversality|Transversality]] of mappings).
  
According to the above definition, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709071.png" /> every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709072.png" /> must be considered as critical. Then, however, there are considerable differences between the properties of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709073.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709074.png" /> and the points for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709075.png" />. In the former case there is a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709076.png" /> in which the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709077.png" /> looks roughly like the standard imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709078.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709079.png" />; more precisely, there exist local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709080.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709081.png" /> (on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709082.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709083.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709084.png" /> (on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709085.png" />), in terms of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709086.png" /> is given by
+
According to the above definition, when $  k < l $
 +
every point $  x _ {0} \in M $
 +
must be considered as critical. Then, however, there are considerable differences between the properties of the points $  x _ {0} $
 +
for which $  \mathop{\rm Rk} _ {x _ {0}  }  f = k $
 +
and the points for which $  \mathop{\rm Rk} _ {x _ {0}  }  f < k $.  
 +
In the former case there is a neighbourhood of $  x _ {0} $
 +
in which the mapping $  f $
 +
looks roughly like the standard imbedding of $  \mathbf R  ^ {k} $
 +
into $  \mathbf R  ^ {l} $;  
 +
more precisely, there exist local coordinates $  x _ {1} \dots x _ {k} $
 +
near $  x _ {0} $(
 +
on $  M $)  
 +
and $  y _ {1} \dots y _ {l} $
 +
near $  f ( x _ {0} ) $(
 +
on $  N $),  
 +
in terms of which $  f $
 +
is given by
  
<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/c/c027/c027090/c02709087.png" /></td> </tr></table>
+
$$
 +
y _ {i}  = x _ {i} ,\ \
 +
i \leq  k; \ \
 +
y _ {k + 1 }  = \dots
 +
= y _ {l}  = 0.
 +
$$
  
In the second case the image of a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709088.png" /> need not be a manifold, displaying instead various singularities — cusps, self-intersections, etc. For this reason, the definition of a critical point is often modified to include only points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709089.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709090.png" />; corresponding modifications are then necessary in the definitions of the other terms listed above [[#References|[5]]].
+
In the second case the image of a neighbourhood of $  x _ {0} $
 +
need not be a manifold, displaying instead various singularities — cusps, self-intersections, etc. For this reason, the definition of a critical point is often modified to include only points $  x _ {0} $
 +
such that $  \mathop{\rm Rk} _ {x _ {0}  }  f < \min ( k, l) $;  
 +
corresponding modifications are then necessary in the definitions of the other terms listed above [[#References|[5]]].
  
The behaviour of mappings in a neighbourhood of a critical point is investigated in the theory of singularities of differentiable mappings (see [[#References|[5]]], [[#References|[6]]]). In that context one studies not arbitrary critical points (concerning which little can be said), but critical points satisfying conditions which ensure that they are "not too strongly degenerate" and "typical" . Thus, one considers critical points of sufficiently smooth mappings, or families of mappings (which depend smoothly on finitely many parameters), which are "unremovable" in the sense that, under small perturbations ( "small" being understood in the sense of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709091.png" /> for suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709092.png" />) of the original mapping, or of the original family, the perturbed mapping (family) has a critical point of the same type in some neighbourhood of the original critical point. For a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027090/c02709093.png" /> (i.e. an ordinary scalar function; in this case critical points are often called stationary points), critical points which are typical in the indicated sense are the non-degenerate critical points at which the Hessian is a non-degenerate quadratic form. Concerning typical critical points for a family of functions see [[#References|[6]]], [[#References|[7]]].
+
The behaviour of mappings in a neighbourhood of a critical point is investigated in the theory of singularities of differentiable mappings (see [[#References|[5]]], [[#References|[6]]]). In that context one studies not arbitrary critical points (concerning which little can be said), but critical points satisfying conditions which ensure that they are "not too strongly degenerate" and "typical" . Thus, one considers critical points of sufficiently smooth mappings, or families of mappings (which depend smoothly on finitely many parameters), which are "unremovable" in the sense that, under small perturbations ( "small" being understood in the sense of $  C  ^ {m} $
 +
for suitable $  m $)  
 +
of the original mapping, or of the original family, the perturbed mapping (family) has a critical point of the same type in some neighbourhood of the original critical point. For a mapping $  M \rightarrow \mathbf R $(
 +
i.e. an ordinary scalar function; in this case critical points are often called stationary points), critical points which are typical in the indicated sense are the non-degenerate critical points at which the Hessian is a non-degenerate quadratic form. Concerning typical critical points for a family of functions see [[#References|[6]]], [[#References|[7]]].
  
 
====References====
 
====References====

Revision as of 17:31, 5 June 2020


For an analytic function $ f ( z) $, a critical point of order $ m $ is a point $ a $ of the complex plane at which $ f ( z) $ is regular but its derivative $ f ^ { \prime } ( z) $ has a zero of order $ m $, where $ m $ is a natural number. In other words, a critical point is defined by the conditions

$$ \lim\limits _ {z \rightarrow a } \frac{f ( z) - f ( a) }{( z - a) ^ {m} } = 0,\ \ \lim\limits _ {z \rightarrow a } \frac{f ( z) - f ( a) }{( z - a) ^ {m+} 1 } \neq 0. $$

A critical point at infinity, $ a = \infty $, of order $ m $, for a function $ f ( z) $ which is regular at infinity, is defined by the conditions

$$ \lim\limits _ {z \rightarrow \infty } [ f ( z) - f ( \infty )] z ^ {m} = 0,\ \ \lim\limits _ {z \rightarrow \infty } [ f ( z) - f ( \infty )] z ^ {m + 1 } \neq 0. $$

Under the analytic mapping $ w = f ( z) $, the angle between two curves emanating from a critical point of order $ m $ is increased by a factor $ m + 1 $. If $ f ( z) $ is regarded as the complex potential of some planar flow of an incompressible liquid, a critical point is characterized by the property that through it pass not one but $ m + 1 $ stream lines, and the velocity of the flow at a critical point vanishes. In terms of the inverse function $ z = \psi ( w) $( i.e. the function for which $ f [ \psi ( w)] \equiv w $), a critical point is an algebraic branch point of order $ m + 1 $.

A point $ a $ of a complex $ ( n - m) $- dimensional irreducible analytic set

$$ M = \ \{ {z \in V } : { f _ {1} ( z) = \dots = f _ {m} ( z) = 0 } \} , $$

the latter being defined in a neighbourhood $ V $ of $ a $ in the complex space $ \mathbf C ^ {n} $ by the conditions $ f _ {1} ( z) = \dots = f _ {m} ( z) = 0 $, where $ f _ {1} \dots f _ {m} $ are holomorphic functions on $ V $ in $ n $ complex variables, $ z = ( z _ {1} \dots z _ {n} ) $, is called a critical point if the rank of the Jacobian matrix $ \| \partial f _ {j} / \partial z _ {k} \| $, $ j = 1 \dots m $, $ k = 1 \dots n $, is less than $ m $. The other points of $ M $ are called regular. There are relatively few critical points on $ M $: They form an analytic set of complex dimension at most $ n - m - 1 $. In particular, when $ m = 1 $, i.e. if $ M = \{ f _ {1} ( z) = 0 \} $, and the dimension of $ M $ is $ n - 1 $, the dimension of the set of critical points is at most $ n - 2 $.

References

[1] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) MR0173749 Zbl 0135.12101
[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

Comments

A point as described under 2) is also called a singular point of $ M $, cf. [a1].

References

[a1] H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) pp. 95 (Translated from German) MR0414912 Zbl 0381.32001

A critical point of a smooth (i.e. continuously differentiable) mapping $ f $ of a $ k $- dimensional differentiable manifold $ M $ into an $ l $- dimensional differentiable manifold $ N $ is a point $ x _ {0} \in M $ such that the rank $ \mathop{\rm Rk} _ {x _ {0} } f $ of $ f $ at this point (i.e. the dimension of the image $ df ( T _ {x _ {0} } M) $ of the tangent space to $ M $ under the differential mapping $ df: T _ {x _ {0} } M \rightarrow T _ {f ( x _ {0} ) } N $) is less than $ l $. The set of all critical points is called the critical set, the image $ f( x _ {0} ) $ of a critical point $ x _ {0} $ is called a critical value, and a point $ y \in N $ which is not the image of any critical point is called a regular point or a regular value (though it need not belong to the image $ f( M) $); non-critical points of $ M $ are also called regular.

According to Sard's theorem, if $ f $ is smooth of class $ C ^ {m} $, $ m > \min ( k - l, 0) $, then the image of the critical set is of the first category in $ N $( i.e. it is the union of at most countably many nowhere-dense sets) and has $ l $- dimensional measure zero (see [1], [2]). The lower bound for $ m $ cannot be weakened (see [3]). The case most frequently needed is $ m = \infty $( in which case the proof is simplified, see [4]). This theorem is widely used for reductions to general position via "small movements" ; for example, it may readily be used to prove that, given two smooth submanifolds in $ \mathbf R ^ {n} $, there exists an arbitrarily small translation of one of them such that their intersection will also be a submanifold (see [2], [4], and also Transversality of mappings).

According to the above definition, when $ k < l $ every point $ x _ {0} \in M $ must be considered as critical. Then, however, there are considerable differences between the properties of the points $ x _ {0} $ for which $ \mathop{\rm Rk} _ {x _ {0} } f = k $ and the points for which $ \mathop{\rm Rk} _ {x _ {0} } f < k $. In the former case there is a neighbourhood of $ x _ {0} $ in which the mapping $ f $ looks roughly like the standard imbedding of $ \mathbf R ^ {k} $ into $ \mathbf R ^ {l} $; more precisely, there exist local coordinates $ x _ {1} \dots x _ {k} $ near $ x _ {0} $( on $ M $) and $ y _ {1} \dots y _ {l} $ near $ f ( x _ {0} ) $( on $ N $), in terms of which $ f $ is given by

$$ y _ {i} = x _ {i} ,\ \ i \leq k; \ \ y _ {k + 1 } = \dots = y _ {l} = 0. $$

In the second case the image of a neighbourhood of $ x _ {0} $ need not be a manifold, displaying instead various singularities — cusps, self-intersections, etc. For this reason, the definition of a critical point is often modified to include only points $ x _ {0} $ such that $ \mathop{\rm Rk} _ {x _ {0} } f < \min ( k, l) $; corresponding modifications are then necessary in the definitions of the other terms listed above [5].

The behaviour of mappings in a neighbourhood of a critical point is investigated in the theory of singularities of differentiable mappings (see [5], [6]). In that context one studies not arbitrary critical points (concerning which little can be said), but critical points satisfying conditions which ensure that they are "not too strongly degenerate" and "typical" . Thus, one considers critical points of sufficiently smooth mappings, or families of mappings (which depend smoothly on finitely many parameters), which are "unremovable" in the sense that, under small perturbations ( "small" being understood in the sense of $ C ^ {m} $ for suitable $ m $) of the original mapping, or of the original family, the perturbed mapping (family) has a critical point of the same type in some neighbourhood of the original critical point. For a mapping $ M \rightarrow \mathbf R $( i.e. an ordinary scalar function; in this case critical points are often called stationary points), critical points which are typical in the indicated sense are the non-degenerate critical points at which the Hessian is a non-degenerate quadratic form. Concerning typical critical points for a family of functions see [6], [7].

References

[1] A. Sard, "The measure of critical values of differentiable maps" Bull. Amer. Math Soc. , 48 (1942) pp. 883–890 MR7523 Zbl 0063.06720
[2] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) MR0193578 Zbl 0129.13102
[3] H. Whitney, "A function not constant on a connected set of critical points" Duke Math. J. , 1 : 4 (1935) pp. 514–517 MR1545896 Zbl 0013.05801 Zbl 61.1117.01 Zbl 61.0262.07
[4] J.W. Milnor, "Topology from the differential viewpoint" , Univ. Virginia Press (1965)
[5] M. Golubitsky, "Stable mappings and their singularities" , Springer (1974) MR0467801 MR0341518 Zbl 0434.58001 Zbl 0429.58004 Zbl 0294.58004
[6] P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) MR0494220 Zbl 0302.58006
[7] V.I. Arnol'd, "Normal forms of functions near degenerate critical points, the Weyl groups , , and Lagrangian singularities" Funktsional. Anal. i Prilozh. , 6 : 4 (1972) pp. 3–25 (In Russian)

D.V. Anosov

How to Cite This Entry:
Critical point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Critical_point&oldid=24408
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article