Strongly Dependent Ordered Abelian Groups and Henselian Fields

TL;DR

This paper characterizes strongly dependent ordered abelian groups with finite dp-rank and applies this to show that strongly dependent fields with henselian valuations preserve strong dependence in their valued structures.

Contribution

It provides a classification of strongly dependent ordered abelian groups and demonstrates that strong dependence is preserved under henselian valuations in fields.

Findings

Strongly dependent ordered abelian groups have finite dp-rank.

Such groups have finite spines and finitely many primes with infinite index.

Strong dependence in fields extends to their henselian valued structures.

Abstract

Strongly dependent ordered abelian groups have finite dp-rank. They are precisely those groups with finite spines and $|\{p\text{ prime}:[G:pG]=\infty\}|<\infty$. We apply this to show that if $K$ is a strongly dependent field, then $(K,v)$ is strongly dependent for any henselian valuation $v$.