Multi-degree bounds on the Betti numbers of real varieties and semi-algebraic sets and applications

TL;DR

This paper introduces refined bounds on the Betti numbers of real algebraic varieties and semi-algebraic sets, unifying various previous results and improving bounds especially for partially quadratic polynomials, with applications to topological complexity.

Contribution

It provides a unified framework for Betti number bounds with multi-degree dependence and significantly improves bounds for partially quadratic polynomials, addressing open problems.

Findings

New bounds on Betti numbers with multi-degree dependence

Unified approach covering various polynomial degree cases

Improved bounds for partially quadratic polynomials

Abstract

We prove new bounds on the Betti numbers of real varieties and semi-algebraic sets that have a more refined dependence on the degrees of the polynomials defining them than results known before. Our method also unifies several different types of results under a single framework, such as bounds depending on the total degrees, on multi-degrees, as well as in the case of quadratic and partially quadratic polynomials. The bounds we present in the case of partially quadratic polynomials offer a significant improvement over what was previously known. Finally, we extend a result of Barone and Basu on bounding the number of connected components of real varieties defined by two polynomials of differing degrees to the sum of all Betti numbers, thus making progress on an open problem posed in that paper.