Stratifications and term description over valued fields with analytic structure, uniform Yomdin-Gromov parametrizations

TL;DR

This paper develops a strong smooth stratification and term description for definable sets over valued fields with analytic structure, enabling uniform parametrizations and desingularization techniques in a non-Archimedean setting.

Contribution

It introduces a new strong stratification and term description framework for valued fields with analytic structure, extending desingularization and parametrization methods.

Findings

Established a strong smooth stratification of definable sets.

Provided a term description of definable functions over valued fields.

Achieved uniform Yomdin-Gromov parametrizations of definable sets.

Abstract

We establish a certain strong smooth stratification of sets and a term description of functions, which are definable over valued fields (possibly non algebraically closed) with analytic structure. The basic tools are: elimination of valued field quantifiers, term structure of definable functions, Lipschitz cell decomposition with preparation of RV-parametrized sets, and a non-Archimedean definable version of Bierstone-Milman's canonical desingularization algorithm, achieved in an earlier paper of ours. As application, uniform Yomdin-Gromov parametrizations of definable sets are given.