Integer-valued polynomials on subsets of quaternion algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915372809060352 |
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| author | Werner, Nicholas J. |
| author_facet | Werner, Nicholas J. |
| contents | Let $R$ be either the ring of Lipschitz quaternions, or the ring of Hurwitz quaternions. Then, $R$ is a subring of the division ring $\mathbb{D}$ of rational quaternions. For $S \subseteq R$, we study the collection $\rm{Int}(S,R) = \{f \in \mathbb{D}[x] \mid f(S) \subseteq R\}$ of polynomials that are integer-valued on $S$. The set $\rm{Int}(S,R)$ is always a left $R$-submodule of $\mathbb{D}[x]$, but need not be a subring of $\mathbb{D}[x]$. We say that $S$ is a ringset of $R$ if $\rm{Int}(S,R)$ is a subring of $\mathbb{D}[x]$. In this paper, we give a complete classification of the finite subsets of $R$ that are ringsets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20609 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integer-valued polynomials on subsets of quaternion algebras Werner, Nicholas J. Rings and Algebras Let $R$ be either the ring of Lipschitz quaternions, or the ring of Hurwitz quaternions. Then, $R$ is a subring of the division ring $\mathbb{D}$ of rational quaternions. For $S \subseteq R$, we study the collection $\rm{Int}(S,R) = \{f \in \mathbb{D}[x] \mid f(S) \subseteq R\}$ of polynomials that are integer-valued on $S$. The set $\rm{Int}(S,R)$ is always a left $R$-submodule of $\mathbb{D}[x]$, but need not be a subring of $\mathbb{D}[x]$. We say that $S$ is a ringset of $R$ if $\rm{Int}(S,R)$ is a subring of $\mathbb{D}[x]$. In this paper, we give a complete classification of the finite subsets of $R$ that are ringsets. |
| title | Integer-valued polynomials on subsets of quaternion algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2412.20609 |