Definable V-topologies, Henselianity and NIP

TL;DR

This paper explores the uniqueness of definable V-topologies in t-henselian NIP fields, linking Shelah's conjecture to henselianity and establishing new conditions for NIP fields with dp-minimal residue fields.

Contribution

It establishes the uniqueness of definable V-topologies on t-henselian NIP fields and connects Shelah's conjecture to the henselianity conjecture for NIP fields.

Findings

At most one definable V-topology exists on a t-henselian NIP field.

In bi-valued NIP fields, henselian and t-henselian valuations are comparable or dependent.

Shelah's conjecture implies the henselianity conjecture for NIP fields, proved for fields with dp-minimal residue fields.

Abstract

We initiate the study of definable V-topolgies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if $(K,v_1,v_2)$ is a bi-valued NIP field with $v_1$ henselian (resp. t-henselian) then $v_1$ and $v_2$ are comparable (resp. dependent). As a consequence Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah's conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is t-henselian.