This paper explores undecidability in theories of positive characteristic function fields in the “geometric” language of rings \(\mathcal {L}_F = \{0,1,+,\times ,F\}\) , where F is a unary predicate for the subset of nonconstant elements of the field. We are motivated by the (still open) question of the decidability of the existential fragment of the \(\mathcal {L}_F\) -theory of \(\mathbb {F}_p(t)\) : a variant on Hilbert’s Tenth Problem for \(\mathbb {F}_p(t)\) . If K denotes the function field of a curve, and has as a constant subfield C an algebraic extension of an odd characteristic finite field (not algebraically closed), we prove the \(\forall ^1\exists\) -fragment of the \(\mathcal {L}_F\) -theory of K is undecidable. We identify an algebraic condition on elements of K that allows existing machinery of Eisenträger and Shlapentokh (used to conclude undecidability of the existential fragment of the theory of K in the language of rings with some constant symbols for elements of \(K \setminus C\) ) to apply to our setting. This work is drawn from the author’s PhD thesis [as reported by Tyrrell (Undecidability in some Field Theories, University of Oxford, Oxford, 2023. https://ora.ox.ac.uk/objects/uuid:3f8d7c47-a54a-4f27-b156-d1116b11b92f)].
