Submission date: 22 September 2022

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 format and 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 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.

Mathematics Subject Classification: 14Gxx, 11Jxx, 03Cxx

Keywords and phrases:

Full text arXiv 2209.10972: pdf, ps.