Diophantine problems over tamely ramified fields

Abstract

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

arXiv comment33 pages. Final version