Non-Archimedean Whitney stratifications

TL;DR

This paper introduces t-stratifications for Henselian valued fields, establishing their existence and demonstrating their ability to encode local and residue field information, with applications to Whitney stratifications and p-adic geometry.

Contribution

It defines and proves the existence of t-stratifications in valued fields, linking them to classical Whitney stratifications and p-adic isometry types, and applies model theory techniques.

Findings

T-stratifications exist for algebraic and analytic subsets in valued fields.

They induce Whitney stratifications in real and complex settings.

They determine ultra-metric isometry types and prove rationality of Poincaré series.

Abstract

We define "t-stratifications", a strong notion of stratifications for Henselian valued fields $K$ of equi-characteristic 0, and prove that they exist. In contrast to classical stratifications in Archimedean fields, t-stratifications also contain non-local information about the stratified sets. For example, they do not only see the singularities in the valued field, but also those in the residue field. Like Whitney stratifications, t-stratifications exist for different classes of subsets of $K^n$, e.g. algebraic subvarieties or certain classes of analytic subsets. The general framework are definable sets (in the sense of model theory) in a language that satisfies certain hypotheses. We give two applications. First, we show that t-stratifications in suitable valued fields $K$ induce classical Whitney stratifications in $\Bbb R$ or $\Bbb C$; in particular, the existence of t-stratifications implies the existence of Whitney stratifications. This uses methods of non-standard analysis. Second, we show how, using t-stratifications, one can determine the ultra-metric isometry type of definable subsets of $\Bbb Z_p^n$ for $p$ sufficiently big. For those $p$, this proves a conjecture stated in a previous article. In particular, this yields a new, geometric proof of the rationality of Poincar\'e series.