Sharply o-minimal structures and sharp cellular decomposition

TL;DR

This paper develops the theory of sharply o-minimal structures, introducing notions of reduction and equivalence, and establishes sharp cell decomposition with bounds on the geometric complexity of definable sets.

Contribution

It introduces the concept of sharp cell decomposition and shows how to reduce any extnormal{ extit{so}}-minimal structure to one with this property, providing bounds on Betti numbers.

Findings

Every extnormal{ extit{so}}-minimal structure can be reduced to one with sharp cell decomposition.

Bounds on Betti numbers of definable sets are established in terms of format and degree.

Different variants of extnormal{ extit{so}}-minimality are shown to be equivalent up to reduction.

Abstract

Sharply o-minimal structures (denoted \so-minimal) are a strict subclass of the o-minimal structures, aimed at capturing some finer features of structures arising from algebraic geometry and Hodge theory. Sharp o-minimality associates to each definable set a pair of integers known as \emph{format} and \emph{degree}, similar to the ambient dimension and degree in the algebraic case; gives bounds on the growth of these quantities under the logical operations; and allows one to control the geometric complexity of a set in terms of its format and degree. These axioms have significant implications on arithmetic properties of definable sets -- for example, \so-minimality was recently used by the authors to settle Wilkie's conjecture on rational points in $\mathbb{R}_{{\exp}}$-definable sets. In this paper we develop some basic theory of sharply o-minimal structures. We introduce the notions of reduction and equivalence on the class of \so-minimal structures. We give three variants of the definition of \so-minimality, of increasing strength, and show that they all agree up to reduction. We also consider the problem of ``sharp cell decomposition'', i.e. cell decomposition with good control on the number of the cells and their formats and degrees. We show that every \so-minimal structure can be reduced to one admitting sharp cell decomposition, and use this to prove bounds on the Betti numbers of definable sets in terms of format and degree.