The existential theory of equicharacteristic henselian valued fields

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 ${\mathbb{F}}_q\left(\left(t\right)\right)$.