  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
$\mathbb{F}_{q}((t))$.
