Real algebraic variety
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) . For each there exists a plane real algebraic curve with this largest number of components — the -curve. (For methods of constructing -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 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 , .
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 . 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 ,
. The following inequality is satisfied for a real algebraic space curve (in ) for even :
(if , this estimate is exact ). 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:
, , . 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 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 , ,
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 ():
cf. the overview .
C) For a regular complete intersection, if is even, is an -variety and the inclusion homomorphism
is zero, then
In this case, if is even, is an -variety and is zero ():
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 ,  for odd . In particular, for a plane real algebraic curve which is an -curve of even order :
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