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Hensel minimality: Geometric criteria for $\ell$-h-minimality

Floris Vermeulen

Abstract

Recently, Cluckers, Halupczok and Rideau-Kikuchi developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this article, we continue the study of Hensel minimality, in particular focusing on $ω$-h-minimality and $\ell$-h-minimality, for $\ell$ a positive integer. Our main results include an analytic criterion for $\ell$-h-minimality, preservation of $\ell$-h-minimality under coarsening of the valuation and $\ell$-dimensional dimensional geometry.

Hensel minimality: Geometric criteria for $\ell$-h-minimality

Abstract

Recently, Cluckers, Halupczok and Rideau-Kikuchi developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this article, we continue the study of Hensel minimality, in particular focusing on -h-minimality and -h-minimality, for a positive integer. Our main results include an analytic criterion for -h-minimality, preservation of -h-minimality under coarsening of the valuation and -dimensional dimensional geometry.
Paper Structure (24 sections, 30 theorems, 97 equations)

This paper contains 24 sections, 30 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline
  4. Acknowledgements
  5. A criterion for $\ell$-h-minimality
  6. Definitions and setting
  7. A criterion for $\ell$-h-minimality
  8. Proof of the equivalence for $\ell=1$
  9. A useful lemma
  10. From $\ell$-h-minimality to (T$\ell$, D)
  11. Preservation under $\mathrm{RV}$-expansion
  12. From (T$\ell$, D) to $\ell$-h-minimality
  13. Hensel minimality in mixed characteristic
  14. Coarsening the valuation
  15. Definitions and basic results
  16. ...and 9 more sections

Key Result

Theorem 2.2.3

Let ${\mathcal{T}}$ be a theory of equicharacteristic zero valued fields in a language ${\mathcal{L}}$ expanding the language of valued fields. Let $\ell\geq 1$ be an integer. Then the following are equivalent.

Theorems & Definitions (68)

  • Definition 2.1.1: CHRV
  • Definition 2.2.1
  • Definition 2.2.2
  • Theorem 2.2.3
  • proof
  • Corollary 2.2.4
  • proof
  • Theorem 2.2.5
  • proof
  • Lemma 2.3.1
  • ...and 58 more