First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability

TL;DR

This paper investigates the definability and undecidability of rings of integral functions over algebraic extensions of function fields, using norm equations and the Hasse Norm Principle, extending prior work to new settings.

Contribution

It introduces new first-order definability results and undecidability proofs for rings of integral functions over algebraic function field extensions, generalizing previous work.

Findings

Definability of integral closure in certain algebraic extensions

Undecidability of first-order theories for these fields and rings

Extension of prior results to broader classes of function fields

Abstract

In this paper, we study questions of definability and decidability for infinite algebraic extensions ${\bf K}$ of $\mathbb{F}_p(t)$ and their subrings of $\mathcal{S}$-integral functions. We focus on fields ${\bf K}$ satisfying a local property which we call $q$-boundedness. This can be considered a function field analogue of prior work of the first author which considered algebraic extensions of $\mathbb{Q}$. One simple consequence of our work states that if ${\bf K}$ is a $q$-bounded Galois extension of $\mathbb{F}_p(t)$, then for infinitely many non-constant $u$ the integral closure $\mathcal{O}_{\bf K}$ of $\mathbb{F}_p[u]$ inside ${\bf K}$ is first-order definable in ${\bf K}$. Under the additional assumption that the constant subfield of ${\bf K}$ is infinite, it follows that both $\mathcal{O}_{\bf K}$ and ${\bf K}$ have undecidable first-order theories, and that $\mathbb{F}_p[w]$ is definable in ${\bf K}$ for every non-constant $w$ in ${\bf K}$. Our primary tools are norm equations and the Hasse Norm Principle, in the spirit of Rumely. Our paper has an intersection with a recent arXiv preprint by Mart\'inez-Ranero, Salcedo, and Utreras, although our definability results are more extensive and undecidability results are much stronger.