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 $t$.

minor changes, to appear in Zeitschrift für Mathematische Logik und Grundlagen der Mathematik