The relative approximation degree in valued function fields

TL;DR

This paper investigates the properties of elements in immediate valued function field extensions, focusing on conditions for henselian rationality and implications for eliminating wild ramification, which are important for local uniformization.

Contribution

It extends Kaplansky's work by identifying when an immediate valued function field of transcendence degree 1 is henselian rational, aiding in the understanding of ramification and uniformization.

Findings

Characterization of elements in immediate valued extensions

Conditions for henselian rationality in transcendence degree 1

Elimination of wild ramification in certain valued function fields

Abstract

We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). If so, then wild ramification can be eliminated in this valued function field. The results presented in this paper are crucial for the first author's proof of henselian rationality over tame fields, which in turn is used in his work on local uniformization.