Distality in valued fields and related structures

TL;DR

This paper explores the concept of distality in valued fields and related structures, providing characterizations, new tools, and examples of distal expansions, advancing understanding in model theory of valued fields.

Contribution

It offers a comprehensive characterization of distality in ordered abelian groups and henselian valued fields, and introduces a relative quantifier elimination method for abelian groups.

Findings

Characterization of distality in a large class of ordered abelian groups

An AKE-style characterization for henselian valued fields

Identification of distal expansions such as the differential field of logarithmic-exponential transseries

Abstract

We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an AKE-style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new tool for analyzing valued fields we employ a relative quantifier elimination for pure short exact sequences of abelian groups.