zbMATH — the first resource for mathematics

On the topology of the zeros of polynomials. (English. Russian original) Zbl 0849.12001
Russ. Acad. Sci., Dokl., Math. 50, No. 1, 90-91 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 337, No. 4, 437-438 (1994).
The author considers a parabolic polynomial \(F \in \mathbb{R} [t, x_2, \dots, x_n]\) of degree \(m\) \(({\partial F \over \partial t} = \Delta F)\). He proves that the number of components of the complement of the zeros of \(F\) is bounded by \(m^{n - 1} + O(m^{n - 2})\). The proof is based on the study of level hypersurfaces associated with parabolic polynomials and on the maximum principle for solutions of a parabolic equation. Several particular cases are discussed.

12E10 Special polynomials in general fields
14J25 Special surfaces
14F45 Topological properties in algebraic geometry