Hensel minimality I

Raf Cluckers (1, 2, 3) , Immanuel Halupczok (4) , Silvain Rideau-Kikuchi (5, 6)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 Université de Lille
3 KU Leuven - Catholic University of Leuven = Katholieke Universiteit Leuven
4 Heinrich Heine Universität Düsseldorf = Heinrich Heine University [Düsseldorf]
5 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
6 CNRS - Centre National de la Recherche Scientifique

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.

Domaines: Logique [math.LO]

Raf Cluckers, Immanuel Halupczok, Silvain Rideau-Kikuchi. Hensel minimality I. Forum of Mathematics, Pi, 2022, ⟨10.1017/fmp.2022.6⟩. ⟨hal-03017168⟩