When does NIP transfer from fields to henselian expansions?

TL;DR

This paper investigates when NIP (not the independence property) transfers from a field to its henselian valued field extensions, showing conditions under which the valued field remains NIP based on valuation definability and residue field properties.

Contribution

It establishes new criteria for NIP transfer to henselian valued fields, especially relating to the definability of valuations and residue field characteristics.

Findings

If the residue field is not separably closed, the valuation is externally definable.

When the residue field is separably closed, the valued field is NIP as a pure valued field.

The definability of the canonical p-henselian valuation plays a key role.

Abstract

Let $K$ be an NIP field and let $v$ be a henselian valuation on $K$. We ask whether $(K,v)$ is NIP as a valued field. By a result of Shelah, we know that if $v$ is externally definable, then $(K,v)$ is NIP. Using the definability of the canonical $p$-henselian valuation, we show that whenever the residue field of $v$ is not separably closed, then $v$ is externally definable. In the case of separably closed residue field, we show that $(K,v)$ is NIP as a pure valued field.