Non-Archimedean Whitney stratifications

Abstract

We define ‘t‐stratifications’, a strong notion of stratifications for Henselian‐valued fields K of equi‐characteristic 0, 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 see those in the residue field. Like Whitney stratifications, t‐stratifications exist for different classes of subsets of Kn, for example, 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 double-struckR or double-struckC; in particular, the existence of t‐stratifications implies the existence of Whitney stratifications. Second, we show how, using t‐stratifications, one can determine the ultra‐metric isometry type of definable subsets of Zpn for p sufficiently big. For those p, this proves a conjecture stated in a previous article. In particular, this yields a geometric explanation of why Poincaré series are rational.

Motivic local density

“…Note that Halupczok [18] has already defined a notion of regular stratification in Henselian fields of equicharacteristic zero, the so-called t-stratifications. Instead of starting from the classical definition of (a f )-stratification as we do, he starts by the property of local trivialization.…”

“…Instead of starting from the classical definition of (a f )-stratification as we do, he starts by the property of local trivialization. This leads to two different notions of regular stratification in this context, and it is unknown whether a common generalization can be found, see [18], open question 9.1.…”

Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

“…It is plausible that the above results will also hold in more general settings of certain tame non-Archimedean geometries considered in the papers [26,27].…”

“…These as yet open problems may be investigated in analytic structures and in the tame non-Archimedean geometries from the papers [26,27] as well. Extending Lipschitz continuous functions f : A → R, with the same Lipschitz constant from a subset A of R n , goes back to McShane and Whitney.…”

Semi‐algebraic triangulation over p‐adically closed fields

“…A different improvement of cell decompositions facing this question is given by stratifications. Such stratifications have been recently introduced in ‐adically closed field , and in more general non‐standard Henselian valued fields . However, their relationship with the ‐adic triangulation is quite unclear at the moment, due to the very peculiar conditions involved in the definition of ‐adic simplexes.…”