First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability
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| Format: | Preprint |
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2024
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| _version_ | 1866929677789036544 |
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| author | Shlapentokh, Alexandra Springer, Caleb |
| author_facet | Shlapentokh, Alexandra Springer, Caleb |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14960 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability Shlapentokh, Alexandra Springer, Caleb Number Theory Logic 11U05 (Primary) 12L05, 11U09 (Secondary) 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. |
| title | First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability |
| topic | Number Theory Logic 11U05 (Primary) 12L05, 11U09 (Secondary) |
| url | https://arxiv.org/abs/2411.14960 |