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

TL;DR

This paper develops geometric criteria for $ ext{ell}$-h-minimality within Hensel minimality, a framework for tame non-Archimedean geometry, extending its applicability and understanding of its properties.

Contribution

It provides an analytic criterion for $ ext{ell}$-h-minimality and studies its stability under valuation coarsening and in $ ext{ell}$-dimensional geometry.

Findings

Established an analytic criterion for $ ext{ell}$-h-minimality.

Proved preservation of $ ext{ell}$-h-minimality under valuation coarsening.

Analyzed $ ext{ell}$-dimensional geometric properties in Hensel 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 $\omega$-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.