Existential fragments of theories of henselian valued fields

TL;DR

This paper investigates specific fragments of the existential theory of henselian valued fields, providing axiomatizations and decidability results for various cases, including fields like _q((t)).

Contribution

It offers new axiomatizations and decidability results for fragments of the existential theory of henselian valued fields with parameters.

Findings

Unconditional axiomatization of the xists_3-theory of _q((t))

Decidability results for various existential fragments in henselian valued fields

Analysis of existential fragments in different characteristic cases

Abstract

We study fragments of the existential theory of henselian valued fields with parameters. This includes the $\exists_n$-fragment in the equicharacteristic or unramified mixed characteristic case, the $\exists_n\exists_1$-fragment in the equicharacteristic case, and the $\exists_n$-fragment in the residue characteristic zero case. For example, we obtain an unconditional axiomatization (and thereby decidability) of the $\exists_3$-theory of $\mathbb{F}_{q}(\!(t)\!)$ in the language of valued fields with a parameter for $t$.