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

In this paper, we study questions of definability and decidability for infinite algebraic extensions $\mathbf{K}$ of ${\mathbb{F}}_p(t)$ and their subrings of $\mathcal{S}$-integral functions. We focus on fields $\mathbf{K}$ 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 $\mathbf{K}$ is a $q$-bounded Galois extension of ${\mathbb{F}}_p(t)$, then for infinitely many non-constant $u$ the integral closure ${\mathcal{O}}_{\mathbf{K}}$ of ${\mathbb{F}}_p[u]$ inside $\mathbf{K}$ is first-order definable in $\mathbf{K}$. Under the additional assumption that the constant subfield of $\mathbf{K}$ is infinite, it follows that both ${\mathcal{O}}_{\mathbf{K}}$ and $\mathbf{K}$ have undecidable first-order theories, and that ${\mathbb{F}}_p[w]$ is definable in $\mathbf{K}$ for every non-constant $w$ in $\mathbf{K}$. 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 Martínez-Ranero, Salcedo, and Utreras, although our definability results are more extensive and undecidability results are much stronger.