Relative model completeness of henselian valued fields with finite ramification and various value groups

TL;DR

This paper studies the model completeness of certain henselian valued fields with finite ramification, identifying conditions under which their theories are model complete in a specific language.

Contribution

It establishes model completeness results for henselian valued fields with finite ramification and particular value group structures, extending prior understanding.

Findings

Model completeness holds when the residue field's theory is model complete.

Results apply to valued fields with finite spines and lexicographic sum value groups.

A one-sorted language suffices for the model completeness results.

Abstract

We investigate the model completeness of the theory of a mixed characteristic henselian valued field with finite ramification relative to the residue field and value group. We address the case in which the valued field has a value group with finite spines, and the case in which the value group is elementarily equivalent to the infinite lexicographic sum of $\mathbb{Z}$ with a minimal positive element. In both cases, we find a one-sorted language in which the theory of the valued field is model complete, if the theory of the residue field is model complete in the language of rings.