The existential theory of equicharacteristic henselian valued fields

TL;DR

This paper establishes that the existential theory of equicharacteristic henselian valued fields is governed by their residue fields, leading to new decidability results, including for fields like _q((t)).

Contribution

It proves an existential Ax-Kochen-Ershov principle for equicharacteristic henselian valued fields, linking their existential theory to that of their residue fields.

Findings

Existential theory of such fields depends only on the residue field.

Unconditional decidability of the existential theory of _q((t)).

The value group does not influence the existential theory.

Abstract

We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax-Kochen-Ershov principle, which roughly says that the existential theory of an equicharacteristic henselian valued field (of arbitrary characteristic) is determined by the existential theory of the residue field; in particular, it is independent of the value group. As an immediate corollary, we get an unconditional proof of the decidability of the existential theory of $\mathbb{F}_{q}((t))$.