Interpolation Domains

Abstract

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.