Non-archimedean stratifications in power bounded $T$-convex fields

TL;DR

This paper demonstrates that functions in power bounded T-convex fields possess the Jacobian property, enabling non-archimedean stratifications and leading to applications in o-minimal structures, with implications for the theory's minimality.

Contribution

It establishes the Jacobian property for definable functions in power bounded T-convex fields and derives non-archimedean stratifications, extending previous work and applying to o-minimal structures.

Findings

Functions have the Jacobian property in power bounded T-convex fields

Existence of non-archimedean stratifications in such fields

T-convex valued fields are b-minimal with centres when T is power bounded

Abstract

We show that functions definable in power bounded $T$-convex fields have the (multidimensional) Jacobian property. Building on work of I. Halupczok, this implies that a certain notion of non-archimedean stratifications is available in such valued fields. From the existence of these stratifications, we derive some applications in an archimedean o-minimal setting. As a minor result, we also show that if $T$ is power bounded, the theory of $T$-convex valued fields is $b$-minimal with centres.