# 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.

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