For an integral domain D with field of fractions K and a subset E ? K, the ring Int(E, D) = {f ∈ K[X] {pipe} f(E) ? D} of integer-valued polynomials on E has been well studied. In this paper we investigate the more general ring Int(r) (E, D) = {f ∈ K[H] {pipe} f~((k))(E) ? D for all 0 ≤ k ≤ r} of integer-valued polynomials and derivatives (up to order r) on the subset {E ? K}. We show that if E is finite and D has the m-generator property, then the ring Intr(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.
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