Definable henselian valuations in positive residue characteristic
Margarete Ketelsen, Simone Ramello, Piotr Szewczyk
TL;DR
The paper advances the understanding of when a non-trivial henselian valuation ring is definable in the ring language by removing the positive-characteristic constraint on the residue field and presenting a precise six-condition criterion tied to the canonical henselian valuation $v_K$ and its residue field $Kv_K$. It leverages divisibility properties, defect theory, and independent defect, combining Beth definability and ultrapower techniques to construct definable valuations from defect data and from violations of divisibility in residue-field extensions. The main theorem provides an exact characterization (with reductions in equicharacteristic zero) and is complemented by concrete examples, including Puiseux-series constructions, to illustrate the necessity and sufficiency of the conditions. The results have implications for model theory of valued fields and Ax–Kochen/Ershov-type analyses by clarifying how residue-characteristic and defect interact with definability in henselian contexts.
Abstract
We study the question of $\mathcal{L}_{\mathrm{ring}}$-definability of non-trivial henselian valuation rings. Building on previous work of Jahnke and Koenigsmann, we provide a characterization of henselian fields that admit a non-trivial definable henselian valuation. In particular, we treat cases where the canonical henselian valuation has positive residue characteristic, using techniques from the model theory and algebra of tame fields.