Model theory of valued fields with an endomorphism

TL;DR

This paper develops a model-theoretic framework for valued fields with endomorphisms, proving relative quantifier elimination and extending results to fields with Frobenius actions in positive characteristic.

Contribution

It introduces a new model-theoretic approach to valued fields with endomorphisms, including the Frobenius, advancing understanding of their logical structure.

Findings

Established relative quantifier elimination for valued fields with endomorphisms

Extended results to Frobenius actions on separably closed valued fields

Built on recent work of Dor and Halevi

Abstract

We establish relative quantifier elimination for valued fields of residue characteristic zero enriched with a non-surjective valued field endomorphism, building on recent work of Dor and Halevi. In particular, we deduce relative quantifier elimination for the limit of the Frobenius action on separably closed valued fields of positive characteristic.