Hensel minimality I

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.

90 pages; changes compared to v2: proof of dimension theory is more self-contained; new section 5.8 about motivic integration

References in corpus (11)

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• Constructible motivic functions and motivic integration
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• Some properties of analytic difference fields
• Hensel minimality I
• Motivic height zeta functions
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Cited by in corpus (6)

• Hensel minimality I
• A Pila--Wilkie theorem for Hensel minimal curves
• Spherically complete models of Hensel minimal valued fields
• Definable valuations on ordered fields
• Tame topology in Hensel minimal structures
• From Cell Decomposition to Motivic Integration, Hensel Minimality, and Point Counting