# Real algebraic variety

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) . 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 , , ; for a generalization of these results to include space curves, see .)

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 , .

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 . 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 $m _ {1}$,

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

. 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 ). 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:

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

, , . In proving this congruence (, ), 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 , ,

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

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

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

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

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 $n$():

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

cf. the overview .

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

$$\chi ( A ) \equiv - \sigma ( \mathbf C A ) + \left \{ \begin{array}{rl} 2 \mathop{\rm mod} 16 & \textrm{ if } d \equiv 2 \mathop{\rm mod} 8 , \\ - 2 \mathop{\rm mod} 16 & \textrm{ if } d \equiv - 2 \mathop{\rm mod} 8 . \\ \end{array} \right .$$

In this case, if $n$ is even, $A$ is an $( M- 2)$- variety and $i _ {*}$ is zero ():

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

$$\chi ( A) \equiv \left \{ \begin{array}{rl} 2 \mathop{\rm mod} 16 & \textrm{ if } m _ {1} \equiv 2 \mathop{\rm mod} 8 , \\ - 2 \mathop{\rm mod} 16 & \textrm{ if } m _ {1} \equiv - 2 \mathop{\rm mod} 8 . \\ \end{array} \right .$$

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

$$\chi ( A) \equiv \left \{ \begin{array}{rl} 0 \mathop{\rm mod} 16 & \textrm{ if } m _ {1} \equiv 0 \mathop{\rm mod} 8, \\ 0 , 4 \mathop{\rm mod} 16 & \textrm{ if } m _ {1} \equiv 2 \mathop{\rm mod} 8 , \\ 0 , - 4 \mathop{\rm mod} 16 & \textrm{ if } m _ {1} \equiv - 2 \mathop{\rm mod} 8 . \\ \end{array} \right .$$

Certain congruences have also been proved ,  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  for real algebraic varieties with singularities. For an interesting approach to the study of real algebraic varieties see .

How to Cite This Entry:
Real algebraic variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Real_algebraic_variety&oldid=49551
This article was adapted from an original article by D.A. Gudkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article