Existentially defining valuations in function fields over large fields

TL;DR

The paper proves that for function fields F of one variable over a large base field K with K[√-1] not algebraically closed, every valuation ring of F containing K is existentially definable in F. The core method extends prior work by using central simple algebras and patching-based local-global principles to define valuation predicates, rather than relying on quadratic forms. As a consequence, the existential theory of such function fields—with appropriately chosen coefficients—is undecidable, linking definability of valuations to Hilbert-type undecidability results. The results cover base fields that are separably closed but not algebraically closed and provide a unified view across known cases, with careful discussion of uniformity and base-field hypotheses.

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.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4
• Lemma 2.5