A Note on Hilbert's "Geometric" Tenth Problem

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.