Nontriviality of rings of integral-valued polynomials

TL;DR

This paper characterizes when the ring of polynomials that take algebraic integer values on a subset of algebraic integers is larger than the usual integer polynomial ring, using topological and algebraic conditions.

Contribution

It provides necessary and sufficient conditions for the nontriviality of rings of integral-valued polynomials on subsets of algebraic integers, involving valuation, topological, and algebraic properties.

Findings

Characterization of nontrivial rings via topological conditions on S.

Conditions involving ramification indices and residue fields.

Use of polynomial closure in algebraic integers.

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 $\overline{\mathbb Z}$.