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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800201.png" /> of real points of an [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800202.png" /> defined over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800203.png" /> of real numbers. A real algebraic variety is said to be non-singular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800204.png" /> is non-singular. In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800205.png" /> is a smooth variety, and its dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800206.png" /> is equal to the dimension of the complex variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800207.png" />; the latter is known as the complexification of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800208.png" />.
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Non-singular regular complete intersections have been most thoroughly studied. These are varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800209.png" /> in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002010.png" /> which are non-singular regular intersections of hypersurfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002013.png" /> is a homogeneous real polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002014.png" /> variables of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002015.png" />. In such a case the matrix
+
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 +
{{TEX|done}}
  
<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/r/r080/r080020/r08002016.png" /></td> </tr></table>
+
The set  $  A = X ( \mathbf R ) $
 +
of real points of an [[Algebraic variety|algebraic variety]]  $  X $
 +
defined over the field  $  \mathbf R $
 +
of real numbers. A real algebraic variety is said to be non-singular if  $  X $
 +
is non-singular. In such a case  $  A $
 +
is a smooth variety, and its dimension  $  \mathop{\rm dim}  A $
 +
is equal to the dimension of the complex variety  $  \mathbf C A = X ( \mathbf C ) $;  
 +
the latter is known as the complexification of the variety  $  A $.
  
has rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002017.png" /> at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002018.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002019.png" />.
+
Non-singular regular complete intersections have been most thoroughly studied. These are varieties  $  X $
 +
in the projective space  $  \mathbf R P  ^ {q} $
 +
which are non-singular regular intersections of hypersurfaces  $  p _ {i} ( z) = 0 $,
 +
$  1 \leq  i \leq  s $,
 +
where  $  p _ {i} ( z) $
 +
is a homogeneous real polynomial in  $  q $
 +
variables of degree  $  m _ {i} $.  
 +
In such a case the matrix
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002020.png" /> denote the real algebraic variety defined as the intersection system
+
$$
 +
\left \|
 +
\frac{\partial  p _ {i} }{\partial  z _ {j} }
 +
\right \|
 +
$$
  
<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/r/r080/r080020/r08002021.png" /></td> </tr></table>
+
has rank  $  s $
 +
at all points  $  z \in \mathbf C A $;  
 +
$  \mathop{\rm dim}  A = n = q- s $.
 +
 
 +
Let  $  B $
 +
denote the real algebraic variety defined as the intersection system
 +
 
 +
$$
 +
p _ {i} ( z)  = 0 ,\  1\leq  i \leq  s- 1,\  p( z)  = p _ {s} ( z) \  \textrm{ and } \ \
 +
= m _ {s} .
 +
$$
  
 
Examples of regular complete intersections are:
 
Examples of regular complete intersections are:
  
1) A plane real algebraic curve; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002025.png" />.
+
1) A plane real algebraic curve; here $  q= 2 $,
 +
$  s= 1 $,
 +
$  \mathbf C B = \mathbf C P  ^ {2} $,
 +
$  B = \mathbf R P  ^ {2} $.
 +
 
 +
2) A real algebraic hypersurface; here  $  s= 1 $,  
 +
$  \mathbf C B = \mathbf C P  ^ {q} $,
 +
$  B = \mathbf R P  ^ {q} $.
 +
In particular, if  $  q= 3 $,
 +
a real algebraic surface is obtained.
 +
 
 +
3) A real algebraic space curve; here  $  q= 3 $,  
 +
$  s= 2 $.
 +
The surface  $  B $
 +
is defined by an equation  $  p _ {1} ( z) = 0 $,
 +
while the curve  $  A $
 +
is cut out on  $  B $
 +
by a surface  $  p _ {2} ( z) = 0 $.
 +
 
 +
A real algebraic curve  $  A $
 +
of order  $  m _ {1} $
 +
in the plane  $  \mathbf R P  ^ {2} $
 +
consists of finitely many components diffeomorphic to a circle. If  $  m _ {1} $
 +
is even, these components are all two-sidedly imbedded in  $  \mathbf R P  ^ {2} $;
 +
if  $  m _ {1} $
 +
is odd, one component is imbedded one-sidedly, while the remaining ones are imbedded two-sidedly. A two-sidedly imbedded component of  $  A $
 +
is called an oval of  $  A $.  
 +
An oval lying inside an odd number of other ovals of  $  A $
 +
is called odd, while the remaining ovals are even.
 +
 
 +
The number of components of a plane real algebraic curve of order  $  m _ {1} $
 +
is not larger than  $  ( m _ {1} - 1 ) ( m _ {1} - 2 ) / 2 + 1 $(
 +
Harnack's theorem) [[#References|[1]]]. For each  $  m _ {1} $
 +
there exists a plane real algebraic curve with this largest number of components — the  $  M $-
 +
curve. (For methods of constructing  $  M $-
 +
curves see [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]; for a generalization of these results to include space curves, see [[#References|[2]]].)
  
2) A real algebraic hypersurface; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002028.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002029.png" />, a real algebraic surface is obtained.
+
D. Hilbert posed in 1900 the problem of studying the topology of real algebraic varieties, and also of the imbedding of real algebraic varieties into  $  \mathbf R P  ^ {q} $
 +
and of one real algebraic variety into another (Hilbert's 16th problem). He also pointed out difficult partial problems: the study of the mutual locations of the ovals of a sixth-order curve, and the topology and the imbedding of a real algebraic fourth-order surface into  $  \mathbf R P  ^ {3} $.
 +
These partial problems have been solved [[#References|[12]]], [[#References|[13]]].
  
3) A real algebraic space curve; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002031.png" />. The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002032.png" /> is defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002033.png" />, while the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002034.png" /> is cut out on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002035.png" /> by a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002036.png" />.
+
For a plane real algebraic curve $  A $
 +
of even order  $  m _ {1} $
 +
the following exact inequality is valid:
  
A real algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002037.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002038.png" /> in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002039.png" /> consists of finitely many components diffeomorphic to a circle. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002040.png" /> is even, these components are all two-sidedly imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002041.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002042.png" /> is odd, one component is imbedded one-sidedly, while the remaining ones are imbedded two-sidedly. A two-sidedly imbedded component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002043.png" /> is called an oval of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002044.png" />. An oval lying inside an odd number of other ovals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002045.png" /> is called odd, while the remaining ovals are even.
+
$$
 +
-  
 +
\frac{1}{8}
 +
( 3 m _ {1}  ^ {2} - 6 m _ {1} )  \leq  P - N  \leq 
 +
\frac{1}{8}
  
The number of components of a plane real algebraic curve of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002046.png" /> is not larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002047.png" /> (Harnack's theorem) [[#References|[1]]]. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002048.png" /> there exists a plane real algebraic curve with this largest number of components — the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002050.png" />-curve. (For methods of constructing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002051.png" />-curves see [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]; for a generalization of these results to include space curves, see [[#References|[2]]].)
+
( 3 m _ {1}  ^ {2} - 6 m _ {1} ) + 1 ,
 +
$$
  
D. Hilbert posed in 1900 the problem of studying the topology of real algebraic varieties, and also of the imbedding of real algebraic varieties into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002052.png" /> and of one real algebraic variety into another (Hilbert's 16th problem). He also pointed out difficult partial problems: the study of the mutual locations of the ovals of a sixth-order curve, and the topology and the imbedding of a real algebraic fourth-order surface into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002053.png" />. These partial problems have been solved [[#References|[12]]], [[#References|[13]]].
+
where  $  P $
 +
is the number of even ovals and $  N $
 +
is the number of odd ovals of $  A $(
 +
Petrovskii's theorem). If  $  m _ {1} $
 +
is odd, a similar inequality is valid for  $  A \cup L $,  
 +
where  $  L $
 +
is a straight line in general position [[#References|[4]]]. When these results are generalized to include the case of a real algebraic hypersurface of even order, the role of the difference  $  P- N $
 +
is played by the [[Euler characteristic|Euler characteristic]] $  \chi ( B _ {+} ) $,
 +
where  $  B _ {+} = \{ {z \in B } : {p( z) \geq  0 } \} $,
 +
while if  $  q $
 +
is odd, the role of  $  P- N $
 +
is played by  $  \chi ( A) $.  
 +
Thus, for a real algebraic hypersurface  $  A $
 +
of even order  $  m _ {1} $,
  
For a plane real algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002054.png" /> of even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002055.png" /> the following exact inequality is valid:
+
$$
 +
| \chi ( B _ {+} ) |  \leq 
 +
\frac{( m _ {1} - 1)  ^ {q} }{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/r/r080/r080020/r08002056.png" /></td> </tr></table>
+
- s( q; m _ {1} ) +
 +
\frac{1}{2}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002057.png" /> is the number of even ovals and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002058.png" /> is the number of odd ovals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002059.png" /> (Petrovskii's theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002060.png" /> is odd, a similar inequality is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002062.png" /> is a straight line in general position [[#References|[4]]]. When these results are generalized to include the case of a real algebraic hypersurface of even order, the role of the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002063.png" /> is played by the [[Euler characteristic|Euler characteristic]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002065.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002066.png" /> is odd, the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002067.png" /> is played by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002068.png" />. Thus, for a real algebraic hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002069.png" /> of even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002070.png" />,
+
where $  s ( q ;  m _ {1)} $
 +
is the number of terms of the polynomial
  
<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/r/r080/r080020/r08002071.png" /></td> </tr></table>
+
$$
 +
\prod _ {i = 1 } ^ { q }  ( 1 + x _ {i} + \dots + x _ {i}  ^ {m-} 2 ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002072.png" /> is the number of terms of the polynomial
+
of degree not higher than  $  ( qm _ {1} - 2q - m _ {1} ) / 2 $;
 +
if  $  q $
 +
is odd, then for any  $  m _ {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/r/r080/r080020/r08002073.png" /></td> </tr></table>
+
$$
 +
| \chi ( A) |  \leq  ( m _ {1} - 1 )  ^ {q} - 2s ( q ; m _ {1} ) + 1 ,
 +
$$
  
of degree not higher than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002074.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002075.png" /> is odd, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002076.png" />,
+
[[#References|[5]]]. The following inequality is satisfied for a real algebraic space curve (in  $  \mathbf R P  ^ {3} $)
 +
for even  $  m _ {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/r/r080/r080020/r08002077.png" /></td> </tr></table>
+
$$
 +
| \chi ( B _ {+} ) |  \leq 
 +
\frac{1}{3}
 +
m _ {1}  ^ {3} +
  
[[#References|[5]]]. The following inequality is satisfied for a real algebraic space curve (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002078.png" />) for even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002079.png" />:
+
\frac{3}{8}
 +
m _ {1} m _ {2}  ^ {2} +
 +
\frac{1}{4}
 +
m _ {1}  ^ {2} m _ {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/r/r080/r080020/r08002080.png" /></td> </tr></table>
+
$$
 +
- m _ {1}  ^ {2} - m _ {1} m _ {2} +
 +
\frac{7}{6}
 +
m _ {1} +
 +
\frac{| \chi ( B) | }{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/r/r080/r080020/r08002081.png" /></td> </tr></table>
+
$$
  
(if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002082.png" />, this estimate is exact [[#References|[6]]]). Petrovskii's theorem has been generalized to arbitrary real algebraic varieties .
+
(if $  m _ {1} = 2 $,  
 +
this estimate is exact [[#References|[6]]]). Petrovskii's theorem has been generalized to arbitrary real algebraic varieties .
  
For a plane real algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002083.png" />-curve of even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002084.png" /> the following congruence is valid:
+
For a plane real algebraic $  M $-
 +
curve of even order $  m _ {1} $
 +
the following congruence is valid:
  
<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/r/r080/r080020/r08002085.png" /></td> </tr></table>
+
$$
 +
P - N  \equiv  \left (
 +
\frac{m _ {1} }{2}
 +
\right )  ^ {2}  \mathop{\rm mod}  8 ,
 +
$$
  
[[#References|[8]]], , [[#References|[13]]]. In proving this congruence ([[#References|[8]]], ), real algebraic varieties were studied by methods of differential topology in a form which opened the way for further investigations. Let the plane real algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002086.png" /> have even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002087.png" /> and let the sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002088.png" /> be chosen so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002089.png" /> is orientable, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002090.png" /> denote, respectively, the number of ovals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002091.png" /> which externally bound the components of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002092.png" /> with positive, zero and negative Euler characteristics. In a similar manner, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002093.png" /> are the numbers of such odd ovals for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002094.png" />. Then [[#References|[8]]], [[#References|[3]]],
+
[[#References|[8]]], , [[#References|[13]]]. In proving this congruence ([[#References|[8]]], ), real algebraic varieties were studied by methods of differential topology in a form which opened the way for further investigations. Let the plane real algebraic curve $  A $
 +
have even order $  m = 2k $
 +
and let the sign of $  p( z) $
 +
be chosen so that $  B _ {+} $
 +
is orientable, while $  P _ {+} , P _ {0} , P _ {-} $
 +
denote, respectively, the number of ovals of $  A $
 +
which externally bound the components of the set $  B _ {+} $
 +
with positive, zero and negative Euler characteristics. In a similar manner, $  N _ {+} , N _ {0} , N _ {-} $
 +
are the numbers of such odd ovals for $  B _ {-} = \{ {z \in B } : {p( z) \leq  0 } \} $.  
 +
Then [[#References|[8]]], [[#References|[3]]],
  
<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/r/r080/r080020/r08002095.png" /></td> </tr></table>
+
$$
 +
P _ {-} + P _ {0}  \leq 
 +
\frac{1}{2}
 +
( k - 1 ) ( k - 2 ) + E ( k) ,
 +
$$
  
<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/r/r080/r080020/r08002096.png" /></td> </tr></table>
+
$$
 +
N _ {-} + N _ {0}  \leq 
 +
\frac{1}{2}
 +
( k - 1 ) ( k - 2 ) ,
 +
$$
  
<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/r/r080/r080020/r08002097.png" /></td> </tr></table>
+
$$
 +
P _ {-}  \geq  N -
 +
\frac{3}{2}
 +
k ( k - 1 ) ,
 +
$$
  
<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/r/r080/r080020/r08002098.png" /></td> </tr></table>
+
$$
 +
N _ {-}  \geq  P -
 +
\frac{3}{2}
 +
k ( k - 1 ) ,
 +
$$
  
 
where
 
where
  
<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/r/r080/r080020/r08002099.png" /></td> </tr></table>
+
$$
 +
E ( k)  =
 +
\frac{1}{2}
 +
( 1 + ( - 1 )  ^ {k} ) .
 +
$$
  
For an arbitrary real algebraic variety in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020100.png" />-dimensional projective space the following inequality is valid:
+
For an arbitrary real algebraic variety in a $  q $-
 +
dimensional projective space the following inequality is valid:
  
<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/r/r080/r080020/r080020101.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dim}  H _ {*} ( A ; \mathbf Z _ {2} )  \leq    \mathop{\rm dim}  H _ {*} ( \mathbf C A ; \mathbf Z _ {2} ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020102.png" /> is the homology space of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020103.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020104.png" /> . This inequality is a generalization of Harnack's theorem. If
+
where $  H _ {*} ( A;  \mathbf Z _ {2} ) = \sum H _ {i} ( A;  \mathbf Z _ {2} ) $
 +
is the homology space of the variety $  A $
 +
with coefficients in $  \mathbf Z _ {2} $.  
 +
This inequality is a generalization of Harnack's theorem. If
  
<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/r/r080/r080020/r080020105.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dim}  H _ {*} ( \mathbf C A ; \mathbf Z _ {2} ) - \mathop{\rm dim}  H _ {*} ( A ; \mathbf Z _ {2} )  = 2t,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020106.png" /> is always an integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020107.png" /> is said to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020108.png" />-variety. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020110.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020111.png" />-variety.
+
where $  t $
 +
is always an integer, $  A $
 +
is said to be an $  ( M- t) $-
 +
variety. If $  t= 0 $,  
 +
$  A $
 +
is an $  M $-
 +
variety.
  
 
The validity of the following congruences has been demonstrated:
 
The validity of the following congruences has been demonstrated:
  
A) For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020112.png" />-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020113.png" /> and for even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020114.png" />:
+
A) For an $  M $-
 +
variety $  A $
 +
and for even $  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/r/r080/r080020/r080020115.png" /></td> </tr></table>
+
$$
 +
\chi ( A )  \equiv  \sigma ( \mathbf C A )  \mathop{\rm mod}  16 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020116.png" /> is the [[Signature|signature]] of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020117.png" /> .
+
where $  \sigma ( \mathbf C A ) $
 +
is the [[Signature|signature]] of the variety $  \mathbf C A $.
  
B) For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020118.png" />-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020119.png" /> and even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020120.png" /> ([[#References|[13]]]):
+
B) For an $  ( M- 1) $-
 +
variety $  A $
 +
and even $  n $([[#References|[13]]]):
  
<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/r/r080/r080020/r080020121.png" /></td> </tr></table>
+
$$
 +
\chi ( A )  \equiv  \sigma ( \mathbf C A ) \pm  2  \mathop{\rm mod}  16 ,
 +
$$
  
 
cf. the overview [[#References|[3]]].
 
cf. the overview [[#References|[3]]].
  
C) For a regular complete intersection, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020122.png" /> is even, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020123.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020124.png" />-variety and the inclusion homomorphism
+
C) For a regular complete intersection, if $  n $
 +
is even, $  A $
 +
is an $  ( M- 1) $-
 +
variety and the inclusion homomorphism
  
<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/r/r080/r080020/r080020125.png" /></td> </tr></table>
+
$$
 +
i _ {*} : H _ {n / 2 }  ( A ; \mathbf Z _ {2} )  \rightarrow  H _ {n / 2 }
 +
( \mathbf R P  ^ {q} ; \mathbf Z _ {2} )
 +
$$
  
 
is zero, then
 
is zero, then
  
<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/r/r080/r080020/r080020126.png" /></td> </tr></table>
+
$$
 +
= m _ {1} m _ {2} \dots  \equiv  2  \mathop{\rm mod}  4
 +
$$
  
 
and
 
and
  
<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/r/r080/r080020/r080020127.png" /></td> </tr></table>
+
$$
 +
\chi ( A )  \equiv  - \sigma ( \mathbf C A ) +
 +
\left \{
  
In this case, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020128.png" /> is even, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020129.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020130.png" />-variety and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020131.png" /> is zero ([[#References|[11]]]):
+
In this case, if $  n $
 +
is even, $  A $
 +
is an $  ( M- 2) $-
 +
variety and $  i _ {*} $
 +
is zero ([[#References|[11]]]):
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020132.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020134.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020135.png" />,
+
if $  d \equiv 0 $
 +
$  \mathop{\rm mod}  8 $,  
 +
$  \chi ( A) \equiv \pm  \sigma ( \mathbf C A ) $
 +
$  \mathop{\rm mod}  16 $,
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020136.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020138.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020139.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020140.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020141.png" />,
+
if $  d \equiv 2 $
 +
$  \mathop{\rm mod}  8 $,  
 +
$  \chi ( A) \equiv - \sigma ( \mathbf C A ) + 4 $
 +
$  \mathop{\rm mod}  16 $
 +
or $  \chi ( A) \equiv \pm  \sigma ( \mathbf C A ) $
 +
$  \mathop{\rm mod}  16 $,
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020142.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020144.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020145.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020146.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020147.png" />.
+
if $  d \begin{array}{c}
 +
> \\
 +
=  
 +
\end{array}
 +
2 $
 +
$  \mathop{\rm mod}  8 $,  
 +
$  \chi ( A) \equiv - \sigma ( \mathbf C A ) - 4 $
 +
$  \mathop{\rm mod}  16 $
 +
or $  \chi ( A) \equiv \pm  \sigma ( \mathbf C A ) $
 +
$  \mathop{\rm mod}  16 $.
  
In particular, for a real algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020148.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020149.png" />,
+
In particular, for a real algebraic surface $  A $
 +
of order $  m _ {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/r/r080/r080020/r080020150.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dim}  H _ {*} ( \mathbf C A ; \mathbf Z _ {2} )  = m _ {1}  ^ {3}
 +
- 4 m _ {1}  ^ {2} + 6 m _ {1} .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020151.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020152.png" />-surface, then
+
If $  A $
 +
is an $  M $-
 +
surface, then
  
<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/r/r080/r080020/r080020153.png" /></td> </tr></table>
+
$$
 +
\chi ( A)  \equiv 
 +
\frac{1}{3}
 +
( 4 m _ {1} - m _ {1}  ^ {3} )  \mathop{\rm mod}  16 .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020154.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020155.png" />-surface, then
+
If $  A $
 +
is an $  ( M- 1) $-
 +
surface, then
  
<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/r/r080/r080020/r080020156.png" /></td> </tr></table>
+
$$
 +
\chi ( A)  \equiv 
 +
\frac{1}{3}
 +
( 4 m _ {1} - m _ {1}  ^ {3} ) \pm  2
 +
  \mathop{\rm mod}  16 .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020157.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020158.png" />-surface and contracts to a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020159.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020160.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020161.png" /> and
+
If $  A $
 +
is an $  ( M- 1) $-
 +
surface and contracts to a point in $  \mathbf R P  ^ {3} $,  
 +
then $  m _ {1} \equiv 2 $
 +
$  \mathop{\rm mod}  4 $
 +
and
  
<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/r/r080/r080020/r080020162.png" /></td> </tr></table>
+
$$
 +
\chi ( A)  \equiv  \left \{
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020163.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020164.png" />-surface and contracts to a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020165.png" />, then
+
If $  A $
 +
is an $  ( M- 2) $-
 +
surface and contracts to a point in $  \mathbf R P  ^ {3} $,  
 +
then
  
<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/r/r080/r080020/r080020166.png" /></td> </tr></table>
+
$$
 +
\chi ( A)  \equiv  \left \{
  
Certain congruences have also been proved , [[#References|[13]]] for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020167.png" />. In particular, for a plane real algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020168.png" /> which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020169.png" />-curve of even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020170.png" />:
+
Certain congruences have also been proved , [[#References|[13]]] for odd $  n $.  
 +
In particular, for a plane real algebraic curve $  A $
 +
which is an $  ( M- 1) $-
 +
curve of even order $  m _ {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/r/r080/r080020/r080020171.png" /></td> </tr></table>
+
$$
 +
P - N  \equiv  \left (
 +
\frac{m _ 1}{2}
 +
\right )  ^ {2}
 +
\pm  1  \mathop{\rm mod}  8 .
 +
$$
  
 
Certain results have also been obtained [[#References|[13]]] for real algebraic varieties with singularities. For an interesting approach to the study of real algebraic varieties see [[#References|[14]]].
 
Certain results have also been obtained [[#References|[13]]] for real algebraic varieties with singularities. For an interesting approach to the study of real algebraic varieties see [[#References|[14]]].
Line 139: Line 381:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Harnack, "Ueber die Vieltheitigkeit der ebenen algebraischen Kurven" ''Math. Ann.'' , '''10''' (1876) pp. 189–198</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Hilbert, "Ueber die reellen Züge algebraischer Kurven" ''Math. Ann.'' , '''38''' (1891) pp. 115–138</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Hilbert, "Mathematische Probleme" ''Arch. Math. Phys.'' , '''1''' (1901) pp. 213–237 {{MR|}} {{ZBL|32.0084.05}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.G. Petrovskii, "On the topology of real plane algebraic curves" ''Ann. of Math.'' , '''39''' : 1 (1938) pp. 189–209 {{MR|1503398}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> O.A. Oleinik, I.G. Petrovskii, "On the topology of real algebraic surfaces" ''Transl. Amer. Math. Soc.'' , '''7''' (1952) pp. 399–417 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''13''' (1949) pp. 389–402 {{MR|0048095}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> O.A. Oleinik, "On the topology of real algebraic curves on an algebraic surface" ''Mat. Sb.'' , '''29''' (1951) pp. 133–156 (In Russian) {{MR|44863}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> , ''Hilbert problems'' , Moscow (1969) (In Russian) {{MR|}} {{ZBL|0187.35502}} {{ZBL|0186.18601}} {{ZBL|0181.15503}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.I. Arnol'd, "Distribution of the ovals of the real plane of algebraic curves, of involutions of four-dimensional smooth manifolds, and the arithmetic of integer-valued quadratic forms" ''Funct. Anal. Appl.'' , '''5''' : 3 (1971) pp. 169–176 ''Funkts. Anal.'' , '''5''' : 3 (1971) pp. 1–9 {{MR|}} {{ZBL|0268.53001}} </TD></TR><TR><TD valign="top">[9a]</TD> <TD valign="top"> V.A. Rokhlin, "Congruences modulo 16 in Hilbert's sixteenth problem" ''Funct. Anal. Appl.'' , '''6''' : 4 (1972) pp. 301–306 ''Funkts. Anal.'' , '''6''' : 4 (1972) pp. 58–64</TD></TR><TR><TD valign="top">[9b]</TD> <TD valign="top"> V.A. Rokhlin, "Congruences modulo 16 in Hilbert's sixteenth problem" ''Funct. Anal. Appl.'' , '''7''' : 2 (1973) pp. 163–165 ''Funkts. Anal.'' , '''7''' : 2 (1973) pp. 91–92</TD></TR><TR><TD valign="top">[10a]</TD> <TD valign="top"> V.M. Kharlamov, "A generalized Petrovskii inequality" ''Funct. Anal. Appl.'' , '''8''' : 2 (1974) pp. 132–137 ''Funkts. Anal.'' , '''8''' : 2 (1974) pp. 50–56 {{MR|}} {{ZBL|0301.14021}} </TD></TR><TR><TD valign="top">[10b]</TD> <TD valign="top"> V.M. Kharlamov, "A generalized Petrovskii inequality II" ''Funct. Anal. Appl.'' , '''9''' : 3 (1975) pp. 266–268 ''Funkts. Anal.'' , '''9''' : 3 (1975) pp. 93–94</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> V.M. Kharlamov, "Additive congruences for the Euler characteristic of real algebraic manifolds of even dimensions" ''Funct. Anal. Appl.'' , '''9''' : 2 (1975) pp. 134–141 ''Funkts. Anal.'' , '''9''' : 2 (1975) pp. 51–60</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> V.M. Kharlamov, "The topological type of nonsingular surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020172.png" /> of degree four" ''Funct. Anal. Appl.'' , '''10''' : 4 (1976) pp. 295–304 ''Funkts. Anal.'' , '''10''' : 4 (1976) pp. 55–68 {{MR|}} {{ZBL|0362.14013}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> D.A. Gudkov, "The topology of real projective algebraic varieties" ''Russian Math. Surveys'' , '''29''' : 4 (1974) pp. 1–80 ''Uspekhi Mat. Nauk'' , '''29''' : 4 (1974) pp. 3–79 {{MR|0399085}} {{ZBL|0316.14018}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> D. Sullivan, "Geometric topology" , '''I. Localization, periodicity, and Galois symmetry''' , M.I.T. (1971) {{MR|0494074}} {{MR|0494075}} {{ZBL|1078.55001}} {{ZBL|0871.57021}} {{ZBL|0366.57003}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Harnack, "Ueber die Vieltheitigkeit der ebenen algebraischen Kurven" ''Math. Ann.'' , '''10''' (1876) pp. 189–198</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Hilbert, "Ueber die reellen Züge algebraischer Kurven" ''Math. Ann.'' , '''38''' (1891) pp. 115–138</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Hilbert, "Mathematische Probleme" ''Arch. Math. Phys.'' , '''1''' (1901) pp. 213–237 {{MR|}} {{ZBL|32.0084.05}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.G. Petrovskii, "On the topology of real plane algebraic curves" ''Ann. of Math.'' , '''39''' : 1 (1938) pp. 189–209 {{MR|1503398}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> O.A. Oleinik, I.G. Petrovskii, "On the topology of real algebraic surfaces" ''Transl. Amer. Math. Soc.'' , '''7''' (1952) pp. 399–417 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''13''' (1949) pp. 389–402 {{MR|0048095}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> O.A. Oleinik, "On the topology of real algebraic curves on an algebraic surface" ''Mat. Sb.'' , '''29''' (1951) pp. 133–156 (In Russian) {{MR|44863}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> , ''Hilbert problems'' , Moscow (1969) (In Russian) {{MR|}} {{ZBL|0187.35502}} {{ZBL|0186.18601}} {{ZBL|0181.15503}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.I. Arnol'd, "Distribution of the ovals of the real plane of algebraic curves, of involutions of four-dimensional smooth manifolds, and the arithmetic of integer-valued quadratic forms" ''Funct. Anal. Appl.'' , '''5''' : 3 (1971) pp. 169–176 ''Funkts. Anal.'' , '''5''' : 3 (1971) pp. 1–9 {{MR|}} {{ZBL|0268.53001}} </TD></TR><TR><TD valign="top">[9a]</TD> <TD valign="top"> V.A. Rokhlin, "Congruences modulo 16 in Hilbert's sixteenth problem" ''Funct. Anal. Appl.'' , '''6''' : 4 (1972) pp. 301–306 ''Funkts. Anal.'' , '''6''' : 4 (1972) pp. 58–64</TD></TR><TR><TD valign="top">[9b]</TD> <TD valign="top"> V.A. Rokhlin, "Congruences modulo 16 in Hilbert's sixteenth problem" ''Funct. Anal. Appl.'' , '''7''' : 2 (1973) pp. 163–165 ''Funkts. Anal.'' , '''7''' : 2 (1973) pp. 91–92</TD></TR><TR><TD valign="top">[10a]</TD> <TD valign="top"> V.M. Kharlamov, "A generalized Petrovskii inequality" ''Funct. Anal. Appl.'' , '''8''' : 2 (1974) pp. 132–137 ''Funkts. Anal.'' , '''8''' : 2 (1974) pp. 50–56 {{MR|}} {{ZBL|0301.14021}} </TD></TR><TR><TD valign="top">[10b]</TD> <TD valign="top"> V.M. Kharlamov, "A generalized Petrovskii inequality II" ''Funct. Anal. Appl.'' , '''9''' : 3 (1975) pp. 266–268 ''Funkts. Anal.'' , '''9''' : 3 (1975) pp. 93–94</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> V.M. Kharlamov, "Additive congruences for the Euler characteristic of real algebraic manifolds of even dimensions" ''Funct. Anal. Appl.'' , '''9''' : 2 (1975) pp. 134–141 ''Funkts. Anal.'' , '''9''' : 2 (1975) pp. 51–60</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> V.M. Kharlamov, "The topological type of nonsingular surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020172.png" /> of degree four" ''Funct. Anal. Appl.'' , '''10''' : 4 (1976) pp. 295–304 ''Funkts. Anal.'' , '''10''' : 4 (1976) pp. 55–68 {{MR|}} {{ZBL|0362.14013}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> D.A. Gudkov, "The topology of real projective algebraic varieties" ''Russian Math. Surveys'' , '''29''' : 4 (1974) pp. 1–80 ''Uspekhi Mat. Nauk'' , '''29''' : 4 (1974) pp. 3–79 {{MR|0399085}} {{ZBL|0316.14018}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> D. Sullivan, "Geometric topology" , '''I. Localization, periodicity, and Galois symmetry''' , M.I.T. (1971) {{MR|0494074}} {{MR|0494075}} {{ZBL|1078.55001}} {{ZBL|0871.57021}} {{ZBL|0366.57003}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> O. Viro, "Successes of the last five years in the topology of real algebraic varieties" , ''Proc. Internat. Congress Mathematicians (Warszawa, 1983)'' , PWN &amp; North-Holland (1984) pp. 603–619</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Wilson, "Hilbert's sixteenth problem" ''Topology'' , '''17''' (1978) pp. 53–74</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> O. Viro, "Successes of the last five years in the topology of real algebraic varieties" , ''Proc. Internat. Congress Mathematicians (Warszawa, 1983)'' , PWN &amp; North-Holland (1984) pp. 603–619</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Wilson, "Hilbert's sixteenth problem" ''Topology'' , '''17''' (1978) pp. 53–74</TD></TR></table>

Revision as of 08:10, 6 June 2020


The set $ A = X ( \mathbf R ) $ of real points of an algebraic variety $ X $ defined over the field $ \mathbf R $ of real numbers. A real algebraic variety is said to be non-singular if $ X $ is non-singular. In such a case $ A $ is a smooth variety, and its dimension $ \mathop{\rm dim} A $ is equal to the dimension of the complex variety $ \mathbf C A = X ( \mathbf C ) $; the latter is known as the complexification of the variety $ A $.

Non-singular regular complete intersections have been most thoroughly studied. These are varieties $ X $ in the projective space $ \mathbf R P ^ {q} $ which are non-singular regular intersections of hypersurfaces $ p _ {i} ( z) = 0 $, $ 1 \leq i \leq s $, where $ p _ {i} ( z) $ is a homogeneous real polynomial in $ q $ variables of degree $ m _ {i} $. In such a case the matrix

$$ \left \| \frac{\partial p _ {i} }{\partial z _ {j} } \right \| $$

has rank $ s $ at all points $ z \in \mathbf C A $; $ \mathop{\rm dim} A = n = q- s $.

Let $ B $ denote the real algebraic variety defined as the intersection system

$$ p _ {i} ( z) = 0 ,\ 1\leq i \leq s- 1,\ p( z) = p _ {s} ( z) \ \textrm{ and } \ \ m = m _ {s} . $$

Examples of regular complete intersections are:

1) A plane real algebraic curve; here $ q= 2 $, $ s= 1 $, $ \mathbf C B = \mathbf C P ^ {2} $, $ B = \mathbf R P ^ {2} $.

2) A real algebraic hypersurface; here $ s= 1 $, $ \mathbf C B = \mathbf C P ^ {q} $, $ B = \mathbf R P ^ {q} $. In particular, if $ q= 3 $, a real algebraic surface is obtained.

3) A real algebraic space curve; here $ q= 3 $, $ s= 2 $. The surface $ B $ is defined by an equation $ p _ {1} ( z) = 0 $, while the curve $ A $ is cut out on $ B $ by a surface $ p _ {2} ( z) = 0 $.

A real algebraic curve $ A $ of order $ m _ {1} $ in the plane $ \mathbf R P ^ {2} $ consists of finitely many components diffeomorphic to a circle. If $ m _ {1} $ is even, these components are all two-sidedly imbedded in $ \mathbf R P ^ {2} $; if $ m _ {1} $ is odd, one component is imbedded one-sidedly, while the remaining ones are imbedded two-sidedly. A two-sidedly imbedded component of $ A $ is called an oval of $ A $. An oval lying inside an odd number of other ovals of $ A $ is called odd, while the remaining ovals are even.

The number of components of a plane real algebraic curve of order $ m _ {1} $ is not larger than $ ( m _ {1} - 1 ) ( m _ {1} - 2 ) / 2 + 1 $( Harnack's theorem) [1]. For each $ m _ {1} $ there exists a plane real algebraic curve with this largest number of components — the $ M $- curve. (For methods of constructing $ M $- curves see [1], [2], [3]; for a generalization of these results to include space curves, see [2].)

D. Hilbert posed in 1900 the problem of studying the topology of real algebraic varieties, and also of the imbedding of real algebraic varieties into $ \mathbf R P ^ {q} $ and of one real algebraic variety into another (Hilbert's 16th problem). He also pointed out difficult partial problems: the study of the mutual locations of the ovals of a sixth-order curve, and the topology and the imbedding of a real algebraic fourth-order surface into $ \mathbf R P ^ {3} $. These partial problems have been solved [12], [13].

For a plane real algebraic curve $ A $ of even order $ m _ {1} $ the following exact inequality is valid:

$$ - \frac{1}{8} ( 3 m _ {1} ^ {2} - 6 m _ {1} ) \leq P - N \leq \frac{1}{8} ( 3 m _ {1} ^ {2} - 6 m _ {1} ) + 1 , $$

where $ P $ is the number of even ovals and $ N $ is the number of odd ovals of $ A $( Petrovskii's theorem). If $ m _ {1} $ is odd, a similar inequality is valid for $ A \cup L $, where $ L $ is a straight line in general position [4]. When these results are generalized to include the case of a real algebraic hypersurface of even order, the role of the difference $ P- N $ is played by the Euler characteristic $ \chi ( B _ {+} ) $, where $ B _ {+} = \{ {z \in B } : {p( z) \geq 0 } \} $, while if $ q $ is odd, the role of $ P- N $ is played by $ \chi ( A) $. Thus, for a real algebraic hypersurface $ A $ of even order $ m _ {1} $,

$$ | \chi ( B _ {+} ) | \leq \frac{( m _ {1} - 1) ^ {q} }{2} - s( q; m _ {1} ) + \frac{1}{2} , $$

where $ s ( q ; m _ {1)} $ is the number of terms of the polynomial

$$ \prod _ {i = 1 } ^ { q } ( 1 + x _ {i} + \dots + x _ {i} ^ {m-} 2 ) , $$

of degree not higher than $ ( qm _ {1} - 2q - m _ {1} ) / 2 $; if $ q $ is odd, then for any $ m _ {1} $,

$$ | \chi ( A) | \leq ( m _ {1} - 1 ) ^ {q} - 2s ( q ; m _ {1} ) + 1 , $$

[5]. The following inequality is satisfied for a real algebraic space curve (in $ \mathbf R P ^ {3} $) for even $ m _ {1} $:

$$ | \chi ( B _ {+} ) | \leq \frac{1}{3} m _ {1} ^ {3} + \frac{3}{8} m _ {1} m _ {2} ^ {2} + \frac{1}{4} m _ {1} ^ {2} m _ {2} + $$

$$ - m _ {1} ^ {2} - m _ {1} m _ {2} + \frac{7}{6} m _ {1} + \frac{| \chi ( B) | }{2} $$

(if $ m _ {1} = 2 $, this estimate is exact [6]). Petrovskii's theorem has been generalized to arbitrary real algebraic varieties .

For a plane real algebraic $ M $- curve of even order $ m _ {1} $ the following congruence is valid:

$$ P - N \equiv \left ( \frac{m _ {1} }{2} \right ) ^ {2} \mathop{\rm mod} 8 , $$

[8], , [13]. In proving this congruence ([8], ), real algebraic varieties were studied by methods of differential topology in a form which opened the way for further investigations. Let the plane real algebraic curve $ A $ have even order $ m = 2k $ and let the sign of $ p( z) $ be chosen so that $ B _ {+} $ is orientable, while $ P _ {+} , P _ {0} , P _ {-} $ denote, respectively, the number of ovals of $ A $ which externally bound the components of the set $ B _ {+} $ with positive, zero and negative Euler characteristics. In a similar manner, $ N _ {+} , N _ {0} , N _ {-} $ are the numbers of such odd ovals for $ B _ {-} = \{ {z \in B } : {p( z) \leq 0 } \} $. Then [8], [3],

$$ P _ {-} + P _ {0} \leq \frac{1}{2} ( k - 1 ) ( k - 2 ) + E ( k) , $$

$$ N _ {-} + N _ {0} \leq \frac{1}{2} ( k - 1 ) ( k - 2 ) , $$

$$ P _ {-} \geq N - \frac{3}{2} k ( k - 1 ) , $$

$$ N _ {-} \geq P - \frac{3}{2} k ( k - 1 ) , $$

where

$$ E ( k) = \frac{1}{2} ( 1 + ( - 1 ) ^ {k} ) . $$

For an arbitrary real algebraic variety in a $ q $- dimensional projective space the following inequality is valid:

$$ \mathop{\rm dim} H _ {*} ( A ; \mathbf Z _ {2} ) \leq \mathop{\rm dim} H _ {*} ( \mathbf C A ; \mathbf Z _ {2} ) , $$

where $ H _ {*} ( A; \mathbf Z _ {2} ) = \sum H _ {i} ( A; \mathbf Z _ {2} ) $ is the homology space of the variety $ A $ with coefficients in $ \mathbf Z _ {2} $. This inequality is a generalization of Harnack's theorem. If

$$ \mathop{\rm dim} H _ {*} ( \mathbf C A ; \mathbf Z _ {2} ) - \mathop{\rm dim} H _ {*} ( A ; \mathbf Z _ {2} ) = 2t, $$

where $ t $ is always an integer, $ A $ is said to be an $ ( M- t) $- variety. If $ t= 0 $, $ A $ is an $ M $- variety.

The validity of the following congruences has been demonstrated:

A) For an $ M $- variety $ A $ and for even $ n $:

$$ \chi ( A ) \equiv \sigma ( \mathbf C A ) \mathop{\rm mod} 16 , $$

where $ \sigma ( \mathbf C A ) $ is the signature of the variety $ \mathbf C A $.

B) For an $ ( M- 1) $- variety $ A $ and even $ n $([13]):

$$ \chi ( A ) \equiv \sigma ( \mathbf C A ) \pm 2 \mathop{\rm mod} 16 , $$

cf. the overview [3].

C) For a regular complete intersection, if $ n $ is even, $ A $ is an $ ( M- 1) $- variety and the inclusion homomorphism

$$ i _ {*} : H _ {n / 2 } ( A ; \mathbf Z _ {2} ) \rightarrow H _ {n / 2 } ( \mathbf R P ^ {q} ; \mathbf Z _ {2} ) $$

is zero, then

$$ d = m _ {1} m _ {2} \dots \equiv 2 \mathop{\rm mod} 4 $$

and

$$ \chi ( A ) \equiv - \sigma ( \mathbf C A ) + \left \{ In this case, if $ n $ is even, $ A $ is an $ ( M- 2) $- variety and $ i _ {*} $ is zero ([[#References|[11]]]): if $ d \equiv 0 $ $ \mathop{\rm mod} 8 $, $ \chi ( A) \equiv \pm \sigma ( \mathbf C A ) $ $ \mathop{\rm mod} 16 $, if $ d \equiv 2 $ $ \mathop{\rm mod} 8 $, $ \chi ( A) \equiv - \sigma ( \mathbf C A ) + 4 $ $ \mathop{\rm mod} 16 $ or $ \chi ( A) \equiv \pm \sigma ( \mathbf C A ) $ $ \mathop{\rm mod} 16 $, if $ d \begin{array}{c} > \\ = \end{array} 2 $ $ \mathop{\rm mod} 8 $, $ \chi ( A) \equiv - \sigma ( \mathbf C A ) - 4 $ $ \mathop{\rm mod} 16 $ or $ \chi ( A) \equiv \pm \sigma ( \mathbf C A ) $ $ \mathop{\rm mod} 16 $. In particular, for a real algebraic surface $ A $ of order $ m _ {1} $, $$

\mathop{\rm dim}  H _ {*} ( \mathbf C A ;  \mathbf Z _ {2} )  =  m _ {1}  ^ {3}

- 4 m _ {1} ^ {2} + 6 m _ {1} . $$ If $ A $ is an $ M $- surface, then $$ \chi ( A) \equiv \frac{1}{3}

( 4 m _ {1} - m _ {1}  ^ {3} )   \mathop{\rm mod}  16 .

$$ If $ A $ is an $ ( M- 1) $- surface, then $$ \chi ( A) \equiv \frac{1}{3}

( 4 m _ {1} - m _ {1}  ^ {3} ) \pm  2
 \mathop{\rm mod}  16 .

$$ If $ A $ is an $ ( M- 1) $- surface and contracts to a point in $ \mathbf R P ^ {3} $, then $ m _ {1} \equiv 2 $ $ \mathop{\rm mod} 4 $ and $$ \chi ( A) \equiv \left \{

If $ A $ is an $ ( M- 2) $- surface and contracts to a point in $ \mathbf R P ^ {3} $, then

$$ \chi ( A) \equiv \left \{ Certain congruences have also been proved , [[#References|[13]]] for odd $ n $. In particular, for a plane real algebraic curve $ A $ which is an $ ( M- 1) $- curve of even order $ m _ {1} $: $$ P - N \equiv \left ( \frac{m _ 1}{2}

\right )  ^ {2}

\pm 1 \mathop{\rm mod} 8 . $$

Certain results have also been obtained [13] for real algebraic varieties with singularities. For an interesting approach to the study of real algebraic varieties see [14].

References

[1] A. Harnack, "Ueber die Vieltheitigkeit der ebenen algebraischen Kurven" Math. Ann. , 10 (1876) pp. 189–198
[2] D. Hilbert, "Ueber die reellen Züge algebraischer Kurven" Math. Ann. , 38 (1891) pp. 115–138
[3] D. Hilbert, "Mathematische Probleme" Arch. Math. Phys. , 1 (1901) pp. 213–237 Zbl 32.0084.05
[4] I.G. Petrovskii, "On the topology of real plane algebraic curves" Ann. of Math. , 39 : 1 (1938) pp. 189–209 MR1503398
[5] O.A. Oleinik, I.G. Petrovskii, "On the topology of real algebraic surfaces" Transl. Amer. Math. Soc. , 7 (1952) pp. 399–417 Izv. Akad. Nauk SSSR Ser. Mat. , 13 (1949) pp. 389–402 MR0048095
[6] O.A. Oleinik, "On the topology of real algebraic curves on an algebraic surface" Mat. Sb. , 29 (1951) pp. 133–156 (In Russian) MR44863
[7] , Hilbert problems , Moscow (1969) (In Russian) Zbl 0187.35502 Zbl 0186.18601 Zbl 0181.15503
[8] V.I. Arnol'd, "Distribution of the ovals of the real plane of algebraic curves, of involutions of four-dimensional smooth manifolds, and the arithmetic of integer-valued quadratic forms" Funct. Anal. Appl. , 5 : 3 (1971) pp. 169–176 Funkts. Anal. , 5 : 3 (1971) pp. 1–9 Zbl 0268.53001
[9a] V.A. Rokhlin, "Congruences modulo 16 in Hilbert's sixteenth problem" Funct. Anal. Appl. , 6 : 4 (1972) pp. 301–306 Funkts. Anal. , 6 : 4 (1972) pp. 58–64
[9b] V.A. Rokhlin, "Congruences modulo 16 in Hilbert's sixteenth problem" Funct. Anal. Appl. , 7 : 2 (1973) pp. 163–165 Funkts. Anal. , 7 : 2 (1973) pp. 91–92
[10a] V.M. Kharlamov, "A generalized Petrovskii inequality" Funct. Anal. Appl. , 8 : 2 (1974) pp. 132–137 Funkts. Anal. , 8 : 2 (1974) pp. 50–56 Zbl 0301.14021
[10b] V.M. Kharlamov, "A generalized Petrovskii inequality II" Funct. Anal. Appl. , 9 : 3 (1975) pp. 266–268 Funkts. Anal. , 9 : 3 (1975) pp. 93–94
[11] V.M. Kharlamov, "Additive congruences for the Euler characteristic of real algebraic manifolds of even dimensions" Funct. Anal. Appl. , 9 : 2 (1975) pp. 134–141 Funkts. Anal. , 9 : 2 (1975) pp. 51–60
[12] V.M. Kharlamov, "The topological type of nonsingular surfaces in of degree four" Funct. Anal. Appl. , 10 : 4 (1976) pp. 295–304 Funkts. Anal. , 10 : 4 (1976) pp. 55–68 Zbl 0362.14013
[13] D.A. Gudkov, "The topology of real projective algebraic varieties" Russian Math. Surveys , 29 : 4 (1974) pp. 1–80 Uspekhi Mat. Nauk , 29 : 4 (1974) pp. 3–79 MR0399085 Zbl 0316.14018
[14] D. Sullivan, "Geometric topology" , I. Localization, periodicity, and Galois symmetry , M.I.T. (1971) MR0494074 MR0494075 Zbl 1078.55001 Zbl 0871.57021 Zbl 0366.57003

Comments

References

[a1] O. Viro, "Successes of the last five years in the topology of real algebraic varieties" , Proc. Internat. Congress Mathematicians (Warszawa, 1983) , PWN & North-Holland (1984) pp. 603–619
[a2] G. Wilson, "Hilbert's sixteenth problem" Topology , 17 (1978) pp. 53–74
How to Cite This Entry:
Real algebraic variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Real_algebraic_variety&oldid=24122
This article was adapted from an original article by D.A. Gudkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article