Quantitative study of semi-Pfaffian sets

TL;DR

This paper provides effective upper bounds on the Betti numbers of semi-Pfaffian sets, advancing understanding of their topological complexity within o-minimal structures.

Contribution

It introduces new bounds on Betti numbers for semi-Pfaffian sets using o-minimal and spectral sequence techniques, extending previous results in tame topology.

Findings

Effective upper bounds on Betti numbers for semi-Pfaffian sets

Extension of topological complexity analysis to o-minimal structures

Application of spectral sequences and Morse theory methods

Abstract

We study the topological complexity of sets defined using Khovanskii's Pfaffian functions, in terms of an appropriate notion of format for those sets. We consider semi- and sub-Pfaffian sets, but more generally any definable set in the o-minimal structure generated by the Pfaffian functions, using the construction of that structure via Gabrielov's notion of limit sets. All the results revolve around giving effective upper-bounds on the Betti numbers (for the singular homology) of those sets. Keywords: Pfaffian functions, fewnomials, o-minimal structures, tame topology, spectral sequences, Morse theory.