A Note on the Decidability of $\mbox{Th}_{\exists}(\mathbb{F}_q (t))$

Brian Tyrrell

Published 2019-09-20Version 1

This note proves that in the language of rings $\mathcal{L}_{\mbox{rings}} = \{+, \cdot, 0, 1\}$ the existential theory of $\mathbb{F}_q (t)$ is decidable, for any finite field $\mathbb{F}_q$ of any characteristic. As a corollary we see $\mbox{Th}_{\exists}(\mathbb{F}_q (t))$ is decidable in the language $\mathcal{L}_{\mbox{rings}} \cup \{F\}$, where $F$ is a unary predicate expressing "$x \not \in \mathbb{F}_q$".