Integer-valued Polynomials

Let R be a Krull ring with quotient field K and a1, . . . , an in R. If and only if the ai are pairwise incongruent mod every height 1 prime ideal of infinite index in R does there exist for all values b1, . . . , bn in R an interpolating integer-valued polynomial, i.e., an f ∈ K[x] with f(ai) = bi and f(R) ⊆ R. If S is an infinite subring of a discrete valuation ring Rv with quotient field K and a1, . . . , an in S are pairwise incongruent mod all M v ∩S of infinite index in S , we derive a formula (depending on the distribution of the ai among residue classes of the ideals M v ∩S) for the minimal d, such that for all b1, . . . , bn ∈Rv there exists a polynomial f ∈K[x] of degree at most d with f(ai) = bi and f(S) ⊆ Rv .