  Assuming resolution of singularities, we prove existential decidability
results for tamely ramified extensions of local fields. In particular, one has
an algorithm to decide whether a variety $V$, defined over $\mathbb{Q}$ (resp.
over $\mathbb{F}_p(t)$), has a rational point over the maximal tamely ramified
extension of $\mathbb{Q}_p$ (resp. of $\mathbb{F}_p((t))$). We also explain how
our results follow from local uniformization.
