Nontriviality of rings of integral-valued polynomials

Abstract: 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 $\overline{\mathbb Z}$.