Nontriviality of rings of integral-valued polynomials
Giulio Peruginelli, Nicholas J. Werner
TL;DR
This work characterizes when rings of integral-valued polynomials $Int_{ ext{Q}}(S,\overline{\mathbb{Z}})$ are nontrivial, connecting global properties of $S$ to local $p$-adic data and valuation-theoretic phenomena. It develops a general framework over valuation domains using pseudo-monotone sequences to give necessary and sufficient conditions for nontriviality, and then translates these criteria to the local $p$-adic setting via the sets $\Sigma_p(S)$ and corresponding rings $Int_{ ext{Q}_p}(\Sigma_p(S),\overline{\mathbb{Z}_p})$. The global analysis employs polynomials like $\Psi_{p,n}$ and Dedekind’s index theorem to construct examples of trivial and nontrivial cases, investigates ramification/residue-field-boundedness as a pathway to nontriviality (and Prüfer properties), and introduces the polynomial closure to characterize when $Int_{ ext{Q}}(S,\overline{\mathbb{Z}})$ remains invariant under enlargement of $S$. The paper also proves that $ ext{Int}_{\mathbb{Q}}(\mathcal{A}_1)=\text{Int}(\mathbb{Z})$ is nontrivial and discusses open questions about polynomial closure for higher degree sets, highlighting deep interactions between arithmetic of algebraic integers and valuation-theoretic structures.
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 $\overline{\mathbb Z}$.