Existential fragments of theories of Henselian valued fields

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 $t$.