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Congruences for real algebraic curves on an ellipsoid. (Russian. English summary) Zbl 0764.14022
The subject of the paper is related to the first part of Hilbert’s 16th problem: topological classification of real algebraic varieties. The result is as follows. Any real curve on the real ellipsoid divides it into two parts with the given curve as their common boundary. The author presents series of new congruences for the Euler characteristic of these parts of the ellipsoid. That gives, in particular, new restrictions to the topology of real curves on the ellipsoid. The most interesting new congruences concern curves of odd bidegrees. It should be noted that they have no analogues for plane curves. The method used consists in applying the Guillou-Marin congruence [cf. L. Guillou and A. Marin, C. R. Acad. Sci., Paris, Sér. A 285, 95-98 (1977; Zbl 0361.57018)] for some \(\mathbb{Z}/4\mathbb{Z}\)-valued quadratic forms defined on one-dimensional homology \(\mathbb{Z}/2\mathbb{Z}\)-classes of a surface embedded into a four- dimensional manifold.

MSC:
14P25 Topology of real algebraic varieties
14H45 Special algebraic curves and curves of low genus
57M50 General geometric structures on low-dimensional manifolds
Citations:
Zbl 0361.57018
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