First-order definability of affine Campana points in the projective line over a number field
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913792912261120 |
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| author | De Rasis, Juan Pablo |
| author_facet | De Rasis, Juan Pablo |
| contents | We offer a $\forall\exists$-definition for (affine) Campana points over $\mathbb{P}^1_K$ (where $K$ is a number field), which constitute a set-theoretical filtration between $K$ and $\mathcal{O}_{K,S}$ ($S$-integers), which are well-known to be universally defined (Koenigsmann 2010, Park 2012, Eisentraeger & Morrison 2016). We also show that our formulas are uniform with respect to all possible $S$, are parameter-free as such, and we count the number of involved quantifiers and offer a bound for the degree of the defining polynomial. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16354 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First-order definability of affine Campana points in the projective line over a number field De Rasis, Juan Pablo Number Theory 11U05 (Primary), 11R04, 11R52 (Secondary) We offer a $\forall\exists$-definition for (affine) Campana points over $\mathbb{P}^1_K$ (where $K$ is a number field), which constitute a set-theoretical filtration between $K$ and $\mathcal{O}_{K,S}$ ($S$-integers), which are well-known to be universally defined (Koenigsmann 2010, Park 2012, Eisentraeger & Morrison 2016). We also show that our formulas are uniform with respect to all possible $S$, are parameter-free as such, and we count the number of involved quantifiers and offer a bound for the degree of the defining polynomial. |
| title | First-order definability of affine Campana points in the projective line over a number field |
| topic | Number Theory 11U05 (Primary), 11R04, 11R52 (Secondary) |
| url | https://arxiv.org/abs/2401.16354 |