  We study the existential theory of equicharacteristic henselian valued fields
with a distinguished uniformizer. In particular, assuming a weak consequence of
resolution of singularities, we obtain an axiomatization of - and therefore an
algorithm to decide - the existential theory relative to the existential theory
of the residue field. This is both more general and works under weaker
resolution hypotheses than the algorithm of Denef and Schoutens, which we also
discuss in detail. In fact, the consequence of resolution of singularities our
results are conditional on is the weakest under which they hold true.
