Integer-Valued Polynomials on a Subset

Abstract LetRbe a Dedekind domain whose residue fields are finite, and letKbe the field of fractions ofR. WhenSis a (non-empty) subset ofKwe write Int(S) for the subring ofK[X] consisting of all polynomialsf(X) inK[X] such thatf(S)⊆R. We show that there exist fractional idealsJ0,J1, …,Jnand monic polynomialsf0,f1, …,fnsuch that Int(S)∩V n = ∑ i=0 n J i f i , n⩾0, whereVnis theK-space of polynomials of degree at mostninK[X]. This generalises classic results on Int(R).