When does NIP transfer from fields to henselian expansions?

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.

8 pages. Contains an unconditional version of the main theorem (even in case the residue field is separably closed)

References in corpus (1)

• On dp-minimal fields

Cited by in corpus (3)

• Characterizing NIP henselian fields
• On Strongly NIP Ordered Fields and Definable Convex Valuations
• Notes on trace equivalence