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

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.