First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability
Alexandra Shlapentokh, Caleb Springer
TL;DR
The paper investigates how definability and decidability questions behave for infinite algebraic extensions ${\bf K}$ of ${\mathbb F}_p(t)$ by introducing the local notion of $q$-boundedness and exploiting norm equations via the Hasse Norm Principle. It proves that for globally $q$-bounded Galois extensions, the integral closure ${\mathcal O}_{\bf K}$ of ${\mathbb F}_p[u]$ inside ${\bf K}$ is first-order definable for infinitely many non-constant $u$, and that with an infinite constant field the theories of ${\bf K}$ and ${\mathcal O}_{\bf K}$ are undecidable while ${\mathbb F}_p[w]$ is definable for every non-constant $w$. The work develops a robust framework linking norm equations, local–global principles, and Diophantine definitions to define valuation rings and rings of ${\mathcal S}$-integers in infinite extensions, and it obtains strong undecidability results under various hypotheses, including connections to recent preprints. A key contribution is a systematic method to obtain existential or first-order definability of valuation rings and ${\mathcal S}$-integers in $q$-bounded extensions, complemented by results on the definability of arbitrary polynomial rings once a single polynomial ring is definable. Taken together, these results illuminate the landscape between decidability/undecidability in function-field-like settings and provide tools for explicit definability in infinite extensions.
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 Martínez-Ranero, Salcedo, and Utreras, although our definability results are more extensive and undecidability results are much stronger.