Minimal Generating Sets for the D-Algebra Int(S, D)

Jacques Boulanger () and Jean-Luc Chabert ()
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Jacques Boulanger: Université de Picardie, LAMFA/CNRS-UMR 7352
Jean-Luc Chabert: Université de Picardie, LAMFA/CNRS-UMR 7352

A chapter in Rings, Polynomials, and Modules, 2017, pp 79-101 from Springer

Abstract: Abstract We are looking for minimal generating sets for the D-algebra Int(S, D) of integer-valued polynomials on any infinite subset S of a Dedekind domain D. For instance, the binomial polynomials X p r , $$\binom{X}{p^{r}},$$ where p is a prime number and r is any nonnegative integer, form a minimal generating set for the classical ℤ $$\mathbb{Z}$$ -algebra Int ( ℤ ) = { f ∈ ℚ [ X ] ∣ f ( ℤ ) ⊆ ℤ } . $$(\mathbb{Z}) =\{ f \in \mathbb{Q}[X]\mid f(\mathbb{Z}) \subseteq \mathbb{Z}\}.$$ In the local case, when D is a valuation domain and S is a regular subset of D, we are able to construct minimal generating sets, but we are not always able to extract from a generating set a minimal one. In particular, we prove that, in local fields, the generating set of integer-valued polynomials obtained by de Shalit and Iceland by means of Lubin-Tate formal group laws is minimal. In our proofs we make an extensive use of Bhargava’s notion of p-ordering.

Keywords: Integer-valued polynomials; Bhargava’s factorials; Minimal generating sets; Lubin-Tate formal group laws; Dirichlet series; Primary: 13F20; Secondary: 11S31; 11B65; 11R42 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-65874-2_5

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