Hensel minimality I

TL;DR

This paper introduces Hensel minimality, a framework for tame geometry on Henselian valued fields, enabling geometric and diophantine applications similar to o-minimality in real geometry.

Contribution

It develops the foundational theory of Hensel minimality, including existence of t-stratifications and Taylor approximation, expanding tame geometry to non-archimedean fields.

Findings

Existence of t-stratifications in Hensel minimal structures

Taylor approximation results for Hensel minimal structures

Applications to Pila-Wilkie point counting and motivic integration

Abstract

We present a framework for tame geometry on Henselian valued fields which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show existence of t-stratifications in Hensel minimal structures and Taylor approximation results which are key to non-archimedean versions of Pila-Wilkie point counting, Yomdin's parameterization results and to motivic integration. In this first paper we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.