Polynomial approximation of Berkovich spaces and definable types

TL;DR

This paper demonstrates that affine Berkovich spaces over maximally complete fields can be approximated by simpler polynomial-based spaces, leading to applications in semi-algebraic sets and model theory.

Contribution

It introduces a polynomial approximation method for Berkovich spaces and connects it to definable types, extending previous results in the field.

Findings

Affine Berkovich spaces can be approximated by polynomial functions of bounded degree.

The approximation yields insights into semi-algebraic sets.

Points in Berkovich spaces correspond to definable types, confirming model-theoretic tameness.

Abstract

We investigate affine Berkovich spaces over maximally complete fields and prove that they may be approximated by simpler spaces when the only functions we need to evaluate are polynomials of bounded degree. We derive applications to semi-algebraic sets and recover a result of E. Hrushovski and F. Loeser which claims that points of Berkovich spaces give rise to definable types (a model-theoretic notion of tameness).