Undecidability of infinite algebraic extensions of $\mathbb{F}_p(t)$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909303160438784 |
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| author | Martinez-Ranero, Carlos Salcedo, Dubraska Utreras, Javier |
| author_facet | Martinez-Ranero, Carlos Salcedo, Dubraska Utreras, Javier |
| contents | Building on work of J. Robinson and A. Shlapentokh, we develop a general framework to obtain definability and decidability results of large classes of infinite algebraic extensions of $\mathbb{F}_p(t)$. As an application, we show that for every odd rational prime $p$ there exist infinitely many primes $r$ such that the fields $\mathbb{F}_{p^a}\left(t^{r^{-\infty}}\right)$ have undecidable first-order theory in the language of rings without parameters. Our method uses character theory to construct families of non-isotrivial elliptic curves whose Mordell-Weil group is finitely generated and of positive rank in $\mathbb{Z}_r$-towers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_01492 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Undecidability of infinite algebraic extensions of $\mathbb{F}_p(t)$ Martinez-Ranero, Carlos Salcedo, Dubraska Utreras, Javier Logic 11U05, 11G05, 03C40, 03D35 Building on work of J. Robinson and A. Shlapentokh, we develop a general framework to obtain definability and decidability results of large classes of infinite algebraic extensions of $\mathbb{F}_p(t)$. As an application, we show that for every odd rational prime $p$ there exist infinitely many primes $r$ such that the fields $\mathbb{F}_{p^a}\left(t^{r^{-\infty}}\right)$ have undecidable first-order theory in the language of rings without parameters. Our method uses character theory to construct families of non-isotrivial elliptic curves whose Mordell-Weil group is finitely generated and of positive rank in $\mathbb{Z}_r$-towers. |
| title | Undecidability of infinite algebraic extensions of $\mathbb{F}_p(t)$ |
| topic | Logic 11U05, 11G05, 03C40, 03D35 |
| url | https://arxiv.org/abs/2409.01492 |