Nontriviality of rings of integral‐valued polynomials

Giulio Peruginelli and Nicholas J. Werner

Mathematische Nachrichten, 2025, vol. 298, issue 12, 3974-3994

Abstract: Let S$S$ be a subset of Z¯$\overline{\mathbb {Z}}$, the ring of all algebraic integers. A polynomial f∈Q[X]$f \in \mathbb {Q}[X]$ is said to be integral‐valued on S$S$ if f(s)∈Z¯$f(s) \in \overline{\mathbb {Z}}$ for all s∈S$s \in S$. The set IntQ(S,Z¯)${\mathrm{Int}}_{\mathbb{Q}}(S,\bar{\mathbb{Z}})$ of all integral‐valued polynomials on S$S$ forms a subring of Q[X]$\mathbb {Q}[X]$ containing Z[X]$\mathbb {Z}[X]$. We say that IntQ(S,Z¯)${\mathrm{Int}} _{\mathbb {Q}}(S,\overline{\mathbb {Z}})$ is trivial if IntQ(S,Z¯)=Z[X]${\mathrm{Int}} _{\mathbb {Q}}(S,\overline{\mathbb {Z}}) = \mathbb {Z}[X]$, and nontrivial otherwise. We give a collection of necessary and sufficient conditions on S$S$ in order for IntQ(S,Z¯)${\mathrm{Int}} _{\mathbb {Q}}(S,\overline{\mathbb {Z}})$ to be nontrivial. Our characterizations involve, variously, topological conditions on S$S$ with respect to fixed extensions of the p$p$‐adic valuations to Q¯$\overline{\mathbb {Q}}$; pseudo‐monotone sequences contained in S$S$; ramification indices and residue field degrees; and the polynomial closure of S$S$ in Z¯$\overline{\mathbb {Z}}$.

Date: 2025
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