Arc-wise analytic t-stratifications

TL;DR

This paper introduces new stratification notions in valued fields, proves their existence and properties, and connects them to Lipschitz stratifications and the Nash-Semple conjecture through a combinatorial invariant.

Contribution

It defines arc-wise analytic t-stratifications, proves their existence, and relates them to t$^2$-stratifications and Lipschitz stratifications, also introducing the critical value function as a new invariant.

Findings

Existence of arc-wise analytic t-stratifications in algebraically closed valued fields.

Arc-wise analytic t-stratifications are t$^2$-stratifications.

t$^2$-stratifications are valuative Lipschitz stratifications.

Abstract

We introduce two new notions of stratifications in valued fields: t$^2$-stratifications and arc-wise analytic t-stratifications. We show the existence of arc-wise analytic t-stratifications in algebraically closed valued fields with analytic structure in the sense of R. Cluckers and L. Lipshitz. We prove that arc-wise analytic t-stratifications are t$^2$-stratifications and, moreover, that t$^2$-stratifications are valuative Lipschitz stratifications as defined by the second author and Y. Yin (the latter ones being closely related to Lipschitz stratifications in the sense of Mostowski). Finally, we introduce a combinatorial invariant associated to a t-stratification which we call the critical value function. We explain how the critical value function of arc-wise analytic t-stratifications can be used to formulate programatic conjectural bounds for the Nash-Semple conjecture.