Diophantine problems over tamely ramified fields

Assuming a certain form of resolution singularities, we prove general existential Ax-Kochen/Ershov principle for tamely ramified fields in all characteristics. This specializes to well-known results residue characteristic 0 and unramified mixed characteristic. It also encompasses the conditional decidability known F p ( t ) its finite extensions, due Denef-Schoutens. On other hand, it applies setting infinite ramification, thereby providing us with an abundance existentially decidable infinitely extensions Q .