Nontriviality of rings of integral-valued polynomials

2024/07/12 by Peruginelli, Giulio, Werner, Nicholas J.
#Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2407.09351

Abstract

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

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