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}$_F = {0, 1, +, \cdot, F}$,withaunarypredicate$F$ for nonconstant elements. In particular we are motivated by a question of Fehm on the decidability of $T{h}_{\mathrm{\exists }}({\mathbb{F}}_p(t);{\mathcal{L}}_F)$; equivalently, that of $T{h}_{\mathrm{\exists }}({\mathbb{F}}_p(t);{\mathcal{L}}_r)$ without parameters. We indicate how to generalise existing machinery to prove the undecidability of $T{h}_{{\mathrm{\forall }}^1\mathrm{\exists }}(K;{\mathcal{L}}_F)$ 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.