Critical points of real polynomials and topology of real algebraic T-surfaces
Résumé
The paper is devoted to a special class of real polynomials, so-called T-polynomials, which arise in the combinatorial version of the Viro theorem. We study the relation between the numbers of real critical points of a given index of a T-polynomial and the combinatorics of lattice triangulations of Newton polytopes. We obtain upper bounds for the numbers of extrema and saddles of generic T-polynomials of a given degree in three variables, and derive from them upper bounds for Betti numbers of real algebraic surfaces in $$\mathbb{R}P^3$$ defined by T-polynomials. The latter upper bounds are stronger than the known upper bounds for arbitrary real algebraic surfaces in $$\mathbb{R}P^3$$ . Another result is the existence of an asymptotically maximal family of real polynomials of degree min three variables with 31m 3/36 + O(m 2) saddle points.