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.
Categories: math.LO math.NT