Definable henselian valuation rings

Abstract

We give model theoretic criteria for $\exists \forall$ and $\forall \exists$- formulas in the ring language to define uniformly the valuation rings $\mathcal{O}$ of models $(K, \mathcal{O})$ of an elementary theory $Σ$ of henselian valued fields. As one of the applications we obtain the existence of an $\exists \forall$-formula defining uniformly the valuation rings $\mathcal{O}$ of valued henselian fields $(K, \mathcal{O})$ whose residue class field $k$ is finite, pseudo-finite, or hilbertian. We also obtain $\forall \exists$-formulas $φ_2$ and $φ_4$ such that $φ_2$ defines uniformly $k[[t]]$ in $k((t))$ whenever $k$ is finite or the function field of a real or complex curve, and $φ_4$ does the job if $k$ is any number field.