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 \(\binom{X}{p^{r}},\) where p is a prime number and r is any nonnegative integer, form a minimal generating set for the classical \(\mathbb{Z}\)-algebra Int\((\mathbb{Z}) =\{ f \in \mathbb{Q}[X]\mid f(\mathbb{Z}) \subseteq \mathbb{Z}\}.\) 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.
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Boulanger, J., Chabert, JL. (2017). Minimal Generating Sets for the D-Algebra Int(S, D). In: Fontana, M., Frisch, S., Glaz, S., Tartarone, F., Zanardo, P. (eds) Rings, Polynomials, and Modules. Springer, Cham. https://doi.org/10.1007/978-3-319-65874-2_5
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