The Skolem property in rings of integer-valued polynomials

Abstract:

Let $D$ be an integral domain with quotient field $K$ and $E\subseteq D$. We investigate the relationship between the Skolem and completely integrally closed properties in the ring of integer-valued polynomials \[ \mathrm {Int}(E, D) = \{f(X) \mid f(X) \in K[X] \text { and } f(a)\in D \text { for every } a\in E\}. \] Among other things, we show for the case $D=\mathbb {Z}$ and $\vert E \vert = \infty$ that the following are equivalent: (1) $\text {Int}(E,\mathbb {Z})$ is strongly Skolem, (2) $\text {Int}(E,\mathbb {Z})$ is completely integrally closed, and (3) $\text {Int}(E,\mathbb {Z})= \text {Int}(E\backslash \{a\}, \mathbb {Z})$ for every $a\in E$.