Artin-Schreier extensions and combinatorial complexity in henselian valued fields

TL;DR

This paper explores the model-theoretic properties of henselian valued fields with Artin-Schreier extensions, providing explicit formulas and extending classification results within NIP and NTP2 contexts.

Contribution

It introduces explicit formulas witnessing IP, IPn, and TP2 in such fields, and generalizes classification results to the NIPn setting for henselian valued fields.

Findings

Explicit formulas witnessing IP, IPn, TP2 in fields with Artin-Schreier extensions

Generalization of NIP classification to NIPn henselian valued fields

NIP henselian valued fields with NIP residue fields are NIP

Abstract

We give explicit formulas witnessing IP, \IPn or TP2 in fields with Artin-Schreier extensions. We use them to control $p$-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the \NIPn context one way of Anscombe-Jahnke's classification of NIP henselian valued fields. As a corollary, we obtain that \NIPn henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.