Existentially defining valuations in function fields over large fields

TL;DR

This paper proves that in certain function fields over large fields, all valuation rings are existentially definable, leading to the undecidability of their existential theories in the language of rings.

Contribution

It extends previous techniques to show all valuation rings in these function fields are existentially definable, implying undecidability of their existential theories.

Findings

All valuation rings containing $K$ are existentially definable.

The existential theory of the function field $F$ is undecidable.

Extension of techniques from earlier work with Becher and Dittmann.

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.