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) [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

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