Existential fragments of theories of henselian valued fields

We study fragments of the existential theory of henselian valued fields with parameters. This includes the ${\mathrm{\exists }}_n$-fragment in the equicharacteristic or unramified mixed characteristic case, the ${\mathrm{\exists }}_n{\mathrm{\exists }}_1$-fragment in the equicharacteristic case, and the ${\mathrm{\exists }}_n$-fragment in the residue characteristic zero case. For example, we obtain an unconditional axiomatization (and thereby decidability) of the ${\mathrm{\exists }}_3$-theory of ${\mathbb{F}}_q(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣})$ in the language of valued fields with a parameter for $t$.