# 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) [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

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

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

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

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 , [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

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