Characterizing diophantine henselian valuation rings and valuation ideals

TL;DR

This paper characterizes diophantine henselian valuation rings and ideals based on their residue fields, generalizing previous results and addressing questions of uniformity and uniqueness in fields.

Contribution

It provides a residue field-based characterization of diophantine henselian valuation rings and ideals, unifying prior findings and establishing bounds on their uniqueness in fields.

Findings

Characterization of diophantine henselian valuation rings via residue fields

Unified framework for positive and negative results in literature

Proof that a field can have at most one diophantine nontrivial henselian valuation

Abstract

We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal.