Diophantine problems over tamely ramified fields

TL;DR

This paper establishes a broad existential principle for tamely ramified fields under certain conditions, unifying known results and extending to infinitely ramified extensions with decidability properties.

Contribution

It proves a general existential Ax-Kochen/Ershov principle for tamely ramified fields, encompassing known and new decidability results in various ramification settings.

Findings

Unified existential principle for tamely ramified fields

Extension of decidability results to infinitely ramified extensions

Specialization to known residue characteristic cases

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.