Integer-valued polynomials on subsets of quaternion algebras

Abstract

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 $\mathrm{Int}(\mathrm{S},\mathrm{R})=\{\mathrm{f}\in \mathbb{D}[\mathrm{x}]\mid \mathrm{f}(\mathrm{S})\subseteq \mathrm{R}\}$ of polynomials that are integer-valued on $S$. The set $\mathrm{Int}(\mathrm{S},\mathrm{R})$ 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 $\mathrm{Int}(\mathrm{S},\mathrm{R})$ is a subring of $\mathbb{D}[x]$. In this paper, we give a complete classification of the finite subsets of $R$ that are ringsets.