A Note on Hilbert's "Geometric" Tenth Problem

TL;DR

This paper investigates the undecidability of certain theories of function fields in positive characteristic, extending existing methods to broader classes of fields and exploring geometric implications.

Contribution

It generalizes machinery to prove undecidability of the universal-existential theory of function fields over algebraic extensions of finite fields.

Findings

Undecidability of $ ext{Th}_{orall^1orall}(K; ext{language})$ for certain function fields.

Extension of undecidability results to non-algebraically closed fields.

Discussion of geometric implications of undecidability in this context.

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.