Recent Progress on Definability of Henselian Valuations

TL;DR

This paper surveys recent advances in the definability of henselian valuations, focusing on concrete cases, existence criteria, and issues of uniformity and quantifier complexity in field theory.

Contribution

It provides a comprehensive overview of recent results on the definability and existence of henselian valuations, highlighting new techniques and unresolved questions.

Findings

Recent results on definability of specific henselian valuations

Criteria for the existence of definable henselian valuations

Discussion of uniformity and quantifier complexity issues

Abstract

Although the study of the definability of henselian valuations has a long history starting with J. Robinson, most of the results in this area were proven during the last few years. We survey these results which address the definability of concrete henselian valuations, the existence of definable henselian valuations on a given field, and questions of uniformity and quantifier complexity.