Effective cylindrical cell decompositions for restricted sub-Pfaffian sets

TL;DR

This paper proves that a modified notion of cell decomposition in restricted sub-Pfaffian sets preserves polynomial complexity in the degree parameter, leading to new polynomial bounds on topological invariants.

Contribution

It introduces a revised notion of format and degree, demonstrating polynomial bounds for cell decomposition and Betti numbers in restricted sub-Pfaffian sets.

Findings

Cell decomposition preserves polynomial dependence on degree D.

First polynomial upper bounds for Betti numbers in this context.

Modified notions of format and degree enable these results.

Abstract

The o-minimal structure generated by the restricted Pfaffian functions, known as restricted sub-Pfaffian sets, admits a natural measure of complexity in terms of a format $\mathcal{F}$, recording information like the number of variables and quantifiers involved in the definition of the set, and a degree $D$ recording the degrees of the equations involved. Khovanskii and later Gabrielov and Vorobjov have established many effective estimates for the geometric complexity of sub-Pfaffian sets in terms of these parameters. It is often important in applications that these estimates are polynomial in $D$. Despite much research done in this area, it is still not known whether cell decomposition, the foundational operation of o-minimal geometry, preserves polynomial dependence on $D$. We slightly modify the usual notions of format and degree and prove that with these revised notions this does in fact hold. As one consequence we also obtain the first polynomial (in $D$) upper bounds for the sum of Betti numbers of sets defined using quantified formulas in the restricted sub-Pfaffian structure.