Definable henselian valuations in positive residue characteristic

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.