Existentially defining valuations in function fields over large fields

Abstract

Let $K$ be a large field such that $K[\sqrt{-1}]$ is not algebraically closed and $F/K$ a function field in one variable. Extending techniques and results from earlier work with Becher and Dittmann, we show that every valuation ring on $F$ containing $K$ is existentially definable in the language of rings with parameters from $F$. As a consequence, using a known reduction technique, we obtain the undecidability of the existential theory of $F$ in the language of rings with appropriately chosen parameters.