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 $D$ of rational quaternions. For $S\subseteq R$, we study the collection $\mathrm{Int}(S,R)=\{f\in D[x\left]\right|f\left(S\right)\subseteq R\}$ of polynomials that are integer-valued on S. The set $\mathrm{Int}(S,R)$ is always a left R-submodule of $D\left[x\right]$, but need not be a subring of $D\left[x\right]$. We say that S is a ringset of R if $\mathrm{Int}(S,R)$ is a subring of $D\left[x\right]$. In this paper, we give a complete classification of the finite subsets of R that are ringsets.