Real critical points of $T$-polynomials that are sums of squared monomials and topology of $T$-hypersurfaces

TL;DR

This paper investigates the topology of certain real algebraic hypersurfaces constructed through combinatorial patchworking, analyzing real critical points of defining polynomials to establish asymptotic bounds on Betti numbers.

Contribution

It introduces new bounds on Betti numbers for hypersurfaces built via dilated triangulations and examines the sharpness of these bounds.

Findings

Asymptotic upper bounds on Betti numbers are established.

The bounds' sharpness is analyzed.

The study links critical points to topological complexity.

Abstract

We study the topology of the real algebraic hypersurfaces in $\mathbb{P}^n$ that can be constructed via combinatorial patchworking using triangulations that are dilations by two of other triangulations. By examining the real critical points of the polynomials that define such hypersurfaces, we find some asymptotical upper bounds on various sums of their Betti numbers. We then discuss the sharpness of those bounds.