Interpolation by Integer-Valued Polynomials

LetRbe a Krull ring with quotient fieldKanda1,…,aninR. If and only if theaiare pairwise incongruent mod every height 1 prime ideal of infinite index inRdoes there exist for all valuesb1,…,bninRan interpolating integer-valued polynomial, i.e., anf K[x] withf(ai) = biandf(R) R.IfSis an infinite subring of a discrete valuation ringRvwith quotient fieldKanda1,…,aninSare pairwise incongruent mod allMkv ∩ Sof infinite index inS, we also determine the minimald(depending on the distribution of theaiamong residue classes of the idealsMkv ∩ S) such that for allb1,…,bn Rvthere exists a polynomialf K[x] of degree at mostdwithf(ai) = biandf(S) Rv.