On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms

TL;DR

This paper establishes new bounds and vanishing results for equivariant Betti numbers of symmetric semi-algebraic sets, introduces a geometric approach, and provides a polynomial-time algorithm for their computation.

Contribution

The paper introduces a novel geometric method to bound and compute equivariant Betti numbers of symmetric semi-algebraic sets, improving previous results and enabling polynomial-time algorithms.

Findings

Vanishing of cohomology groups in high dimensions for fixed degree

Upper bounds on equivariant Betti numbers that are tight up to a degree-dependent factor

Polynomial-time algorithm for computing equivariant Betti numbers

Abstract

Let $\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in dimensions $d$ and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of $d^{O(d)} s^d k^{\lfloor d/2 \rfloor-1} $ on the equivariant Betti numbers of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by quantifier-free formulas involving $s$ symmetric polynomials of degrees bounded by $d$, where $1 < d \ll s,k$. This bound is tight up to a factor depending only on $d$. These results significantly improve upon those obtained previously which were proved using different techniques. Our new methods are quite general, and also yield bounds on the equivariant Betti numbers of certain special classes of symmetric definable sets (definable sets symmetrized by pulling back under symmetric polynomial maps of fixed degree) in arbitrary o-minimal structures over $\mathrm{R}$. Finally, we utilize our new approach to obtain an algorithm with polynomially bounded complexity for computing these equivariant Betti numbers. In contrast, the problem of computing the ordinary Betti numbers of (not necessarily symmetric) semi-algebraic sets is considered to be an intractable problem, and all known algorithms for this problem have doubly exponential complexity.