Nontriviality of rings of integral-valued polynomials
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910044222652416 |
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| author | Peruginelli, Giulio Werner, Nicholas J. |
| author_facet | Peruginelli, Giulio Werner, Nicholas J. |
| contents | Let $S$ be a subset of $\overline{\mathbb Z}$, the ring of all algebraic integers. A polynomial $f \in \mathbb Q[X]$ is said to be integral-valued on $S$ if $f(s) \in \overline{\mathbb Z}$ for all $s \in S$. The set $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z})$ of all integral-valued polynomials on $S$ forms a subring of $\mathbb Q[X]$ containing $\mathbb Z[X]$. We say that $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z})$ is trivial if $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z}) = \mathbb Z[X]$, and nontrivial otherwise. We give a collection of necessary and sufficient conditions on $S$ in order $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z})$ to be nontrivial. Our characterizations involve, variously, topological conditions on $S$ with respect to fixed extensions of the $p$-adic valuations to $\overline{\mathbb Q}$; pseudo-monotone sequences contained in $S$; ramification indices and residue field degrees; and the polynomial closure of $S$ in $\overline{\mathbb Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nontriviality of rings of integral-valued polynomials Peruginelli, Giulio Werner, Nicholas J. Number Theory Commutative Algebra Let $S$ be a subset of $\overline{\mathbb Z}$, the ring of all algebraic integers. A polynomial $f \in \mathbb Q[X]$ is said to be integral-valued on $S$ if $f(s) \in \overline{\mathbb Z}$ for all $s \in S$. The set $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z})$ of all integral-valued polynomials on $S$ forms a subring of $\mathbb Q[X]$ containing $\mathbb Z[X]$. We say that $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z})$ is trivial if $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z}) = \mathbb Z[X]$, and nontrivial otherwise. We give a collection of necessary and sufficient conditions on $S$ in order $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z})$ to be nontrivial. Our characterizations involve, variously, topological conditions on $S$ with respect to fixed extensions of the $p$-adic valuations to $\overline{\mathbb Q}$; pseudo-monotone sequences contained in $S$; ramification indices and residue field degrees; and the polynomial closure of $S$ in $\overline{\mathbb Z}$. |
| title | Nontriviality of rings of integral-valued polynomials |
| topic | Number Theory Commutative Algebra |
| url | https://arxiv.org/abs/2407.09351 |