Non-Archimedean Whitney stratifications

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 $\mathbb{R}$ or $\mathbb{C}$; 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 ${\mathbb{Z}}_p^n$ 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 Poincaré series.

Fixed typos; enhanced the presentation

References in corpus (2)

• Constructible motivic functions and motivic integration
• Trees of definable sets over the p-adics

Cited by in corpus (4)

• Hensel minimality I
• Motivic local density
• Definable transformation to normal crossings over Henselian fields with separated analytic structure
• Semi-algebraic triangulation over p-adically closed fields