A Note on Hilbert's "Geometric" Tenth Problem

Abstract

This paper explores undecidability in theories of positive characteristic function fields in the "geometric" language of rings ${\mathcal{L}}_F=\{0,1,+,\cdot ,F\}$, with a unary predicate $F$ for nonconstant elements. In particular we are motivated by a question of Fehm on the decidability of $\mbox{Th}_{\exists}(\mathbb{F}_p(t); \mathcal{L}_F)$; equivalently, that of $\mbox{Th}_{\exists}(\mathbb{F}_p(t); \mathcal{L}_r)$ without parameters. We indicate how to generalise existing machinery to prove the undecidability of $\mbox{Th}_{\forall^1\exists}(K; \mathcal{L}_F)$ without parameters, where $K$ is the function field of a curve over an algebraic extension of ${\mathbb{F}}_p$, not algebraically closed. We discuss the problem (and its geometric implications) further in this context too.

14 pages, comments welcome. Corrected result from previous version