VC density of definable families over valued fields

TL;DR

This paper establishes an optimal bound on the VC density of definable families over algebraically closed valued fields, extending previous results to all characteristics and connecting to Betti number bounds in Berkovich spaces.

Contribution

It provides the first optimal VC density bound for definable families in algebraically closed valued fields without characteristic restrictions.

Findings

Bound on VC density is optimal and characteristic-independent.

Established new bounds on Betti numbers of semi-algebraic sets in Berkovich spaces.

Connected model theory with topological bounds in non-archimedean geometry.

Abstract

We prove a tight bound on the number of realized $0/1$ patterns (or equivalently on the Vapnik-Chervonenkis codensity) of definable families in models of the theory of algebraically closed valued fields with a non-archimedean valuation. Our result improves the best known result in this direction proved by Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko, who proved a weaker bound in the restricted case where the characteristics of the field $K$ and its residue field are both assumed to be $0$. The bound obtained here is optimal and without any restriction on the characteristics. We obtain the aforementioned bound as a consequence of another result on bounding the Betti numbers of semi-algebraic subsets of certain Berkovich analytic spaces, mirroring similar results known already in the case of o-minimal structures and for real closed, as well as, algebraically closed fields. The latter result is the first result in this direction and is possibly of independent interest. Its proof relies heavily on recent results of Hrushovski and Loeser on the topology of semi-algebraic subsets of Berkovich analytic spaces.