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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" />.
r0800201.png
 
$#A+1 = 170 n = 1
 
$#C+1 = 170 : ~/encyclopedia/old_files/data/R080/R.0800020 Real algebraic variety
 
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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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The set  $  A = X ( \mathbf R ) $
+
<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>
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 $.
 
  
Non-singular regular complete intersections have been most thoroughly studied. These are varieties  $  X $
+
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" />.
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 \|
 
$$
 
  
has rank  $  s $
+
<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>
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:
 
Examples of regular complete intersections are:
  
1) A plane real algebraic curve; here $  q= 2 $,
+
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" />.
$  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]]].)
 
  
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} $
+
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.
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]]].
 
  
For a plane real algebraic curve $  A $
+
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" />.
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}
 
  
( 3 m _ {1}  ^ {2} - 6 m _ {1} ) + 1 ,
+
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]]].)
$$
 
  
where  $  P $
+
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]]].
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}
 
  
- s( q; 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/r08002056.png" /></td> </tr></table>
\frac{1}{2}
 
,
 
$$
 
  
where s ( q ;  m _ {1)} $
+
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" />,
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 ) ,
 
$$
 
  
of degree not higher than  $  ( qm _ {1} - 2q - m _ {1} ) / 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
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 ,
 
$$
 
  
[[#References|[5]]]. The following inequality is satisfied for a real algebraic space curve (in  $  \mathbf R P  ^ {3} $)
+
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" />,
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} +
 
  
\frac{3}{8}
+
[[#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" />:
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 $  m _ {1} = 2 $,  
+
(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 .
this estimate is exact [[#References|[6]]]). Petrovskii's theorem has been generalized to arbitrary real algebraic varieties .
 
  
For a plane real algebraic $  M $-
+
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:
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 $  A $
+
[[#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]]],
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 $  q $-
+
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:
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 $  H _ {*} ( A;  \mathbf Z _ {2} ) = \sum H _ {i} ( 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
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 $  t $
+
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.
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 $  M $-
+
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" />:
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 $  \sigma ( \mathbf C A ) $
+
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" /> .
is the [[Signature|signature]] of the variety $  \mathbf C A $.
 
  
B) For an $  ( M- 1) $-
+
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]]]):
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 $  n $
+
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
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 $  n $
+
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]]]):
is even, $  A $
 
is an $  ( M- 2) $-
 
variety and $  i _ {*} $
 
is zero ([[#References|[11]]]):
 
  
if $  d \equiv 0 $
+
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" />,
$  \mathop{\rm mod}  8 $,  
 
$  \chi ( A) \equiv \pm  \sigma ( \mathbf C A ) $
 
$  \mathop{\rm mod}  16 $,
 
  
if $  d \equiv 2 $
+
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" />,
$  \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}
+
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" />.
> \\
 
=  
 
\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 $
+
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" />,
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 $  A $
+
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
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 $  A $
+
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
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 $  A $
+
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
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 $  A $
+
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
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 $  n $.  
+
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" />:
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 381: Line 139:
 
====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>
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====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 14:53, 7 June 2020

The set of real points of an algebraic variety defined over the field of real numbers. A real algebraic variety is said to be non-singular if is non-singular. In such a case is a smooth variety, and its dimension is equal to the dimension of the complex variety ; the latter is known as the complexification of the variety .

Non-singular regular complete intersections have been most thoroughly studied. These are varieties in the projective space which are non-singular regular intersections of hypersurfaces , , where is a homogeneous real polynomial in variables of degree . In such a case the matrix

has rank at all points ; .

Let denote the real algebraic variety defined as the intersection system

Examples of regular complete intersections are:

1) A plane real algebraic curve; here , , , .

2) A real algebraic hypersurface; here , , . In particular, if , a real algebraic surface is obtained.

3) A real algebraic space curve; here , . The surface is defined by an equation , while the curve is cut out on by a surface .

A real algebraic curve of order in the plane consists of finitely many components diffeomorphic to a circle. If is even, these components are all two-sidedly imbedded in ; if is odd, one component is imbedded one-sidedly, while the remaining ones are imbedded two-sidedly. A two-sidedly imbedded component of is called an oval of . An oval lying inside an odd number of other ovals of is called odd, while the remaining ovals are even.

The number of components of a plane real algebraic curve of order is not larger than (Harnack's theorem) [1]. For each there exists a plane real algebraic curve with this largest number of components — the -curve. (For methods of constructing -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 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 . These partial problems have been solved [12], [13].

For a plane real algebraic curve of even order the following exact inequality is valid:

where is the number of even ovals and is the number of odd ovals of (Petrovskii's theorem). If is odd, a similar inequality is valid for , where 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 is played by the Euler characteristic , where , while if is odd, the role of is played by . Thus, for a real algebraic hypersurface of even order ,

where is the number of terms of the polynomial

of degree not higher than ; if is odd, then for any ,

[5]. The following inequality is satisfied for a real algebraic space curve (in ) for even :

(if , this estimate is exact [6]). Petrovskii's theorem has been generalized to arbitrary real algebraic varieties .

For a plane real algebraic -curve of even order the following congruence is valid:

[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 have even order and let the sign of be chosen so that is orientable, while denote, respectively, the number of ovals of which externally bound the components of the set with positive, zero and negative Euler characteristics. In a similar manner, are the numbers of such odd ovals for . Then [8], [3],

where

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

where is the homology space of the variety with coefficients in . This inequality is a generalization of Harnack's theorem. If

where is always an integer, is said to be an -variety. If , is an -variety.

The validity of the following congruences has been demonstrated:

A) For an -variety and for even :

where is the signature of the variety .

B) For an -variety and even ([13]):

cf. the overview [3].

C) For a regular complete intersection, if is even, is an -variety and the inclusion homomorphism

is zero, then

and

In this case, if is even, is an -variety and is zero ([11]):

if , ,

if , or ,

if , or .

In particular, for a real algebraic surface of order ,

If is an -surface, then

If is an -surface, then

If is an -surface and contracts to a point in , then and

If is an -surface and contracts to a point in , then

Certain congruences have also been proved , [13] for odd . In particular, for a plane real algebraic curve which is an -curve of even order :

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=48448
This article was adapted from an original article by D.A. Gudkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article