AKE principles for deeply ramified fields

TL;DR

This paper explores the model theory of deeply ramified fields of positive characteristic, extending known principles from perfectoid fields to a broader class of ramified fields.

Contribution

It generalizes Ax-Kochen/Ershov principles to certain deeply ramified fields of positive characteristic with fixed imperfection degree.

Findings

Established Ax-Kochen/Ershov principles for these fields

Applied results to all deeply ramified henselian valued fields of rank 1

Extended model-theoretic understanding of ramified fields

Abstract

We study the model theory of deeply ramified fields of positive characteristic. Generalizing the perfect case treated in work by Jahnke and Kartas on the model theory of perfectoid fields, we obtain Ax-Kochen/Ershov principles for certain deeply ramified fields of positive characteristic and fixed degree of imperfection. Our results apply in particular to all deeply ramified henselian valued fields of rank 1.