Interpolation Domains

Call a domain D with quotient field K an interpolation domain if, for each choice of distinct arguments a1,…,an and arbitrary values c1,…,cn in D, there exists an integer-valued polynomial f (that is, f K[X] with f(D) (D)), such that f(ai) = ci for 1 ≤ i ≤ n. We characterize completely the interpolation domains if D is Noetherian or a Prüfer domain. In the first case, we show that D is an interpolation domain if and only if it is one-dimensional, locally unibranched with finite residue fields, in the second one, if and only if the ring Int(D) = {f K[X]f(D) D} of integer-valued polynomials is itself a Prüfer domain. We also show that an interpolation domain must satisfy a double-boundedness condition, and thereby simplify a recent characterization of the domains D such that Int(D) is a Prüfer domain.