Lipschitz stratifications in power-bounded o-minimal fields

TL;DR

This paper introduces a non-archimedean approach to establish Lipschitz stratifications for closed definable sets in power-bounded o-minimal structures, avoiding traditional resolution techniques and providing uniform stratifications.

Contribution

It presents a novel non-archimedean method to prove Lipschitz stratifications, bypassing resolution of singularities and Weierstrass preparation, and achieves uniform stratifications for families of sets.

Findings

Lipschitz stratifications exist in power-bounded o-minimal structures.

The non-archimedean approach sharpens estimates into valuation inequalities.

Uniform stratifications are obtained for families of definable sets.

Abstract

We propose to grok Lipschitz stratifications from a non-archimedean point of view and thereby show that they exist for closed definable sets in any power-bounded o-minimal structure on a real closed field. Unlike the previous approaches in the literature, our method bypasses resolution of singularities and Weierstrass preparation altogether; it transfers the situation to a non-archimedean model, where the quantitative estimates appearing in Lipschitz stratifications are sharpened into valuation-theoretic inequalities. Applied to a uniform family of sets, this approach automatically yields a family of stratifications which satisfy the Lipschitz conditions in a uniform way.