Rings of integer-valued polynomials and derivatives on finite sets

Abstract

For an integral domain D with field of fractions K and a subset \({E \subseteq K}\) , the ring \({{\rm Int}(E, D) = \{f \in K[X]\, |\, f(E) \subseteq D \}}\) of integer-valued polynomials on E has been well studied. In this paper we investigate the more general ring \({{\rm Int}^{(r)}(E, D) = \{ f \in K[X] \, |\, f^{(k)}(E) \subseteq D \,{\rm for\, all}\, 0 \leq k \leq r \}}\) of integer-valued polynomials and derivatives (up to order r) on the subset \({E \subseteq K}\) . We show that if E is finite and D has the m-generator property, then the ring \({{\rm Int}^{(r)}(E, D)}\) has the (r + 1)m-generator property, provided r ≥ 1 or m ≥ 2. We also construct an example to show that this is, in general, the best bound possible.