J. Algebra 211 (1999) 562-577

Interpolation by Integer-Valued Polynomials

Abstract: Let R be a Krull ring with quotient field K and a_1,...,a_n distinct elements of R. If and only if the a_i are pairwise incongruent mod every height 1 prime ideal of infinite index in R does there exist for every choice of b_1,..,b_n in R an interpolating integer-valued polynomial, i.e., f in K[x] such that f(a_i)=b_i and f(R) is contained in R.

In particular, interpolation by integer-valued polynomials is possible for arbitrary arguments and values in a Dedekind ring all of whose residue fields are finite, such as the ring of algebraic integers in a number field.

If R is an infinite subring of a discrete valuation ring with finite residue field and a_1,..,a_n in R, we also determine the minimal d such that for all b_1,..,b_n in R_v there exists f in K[x] of degree at most d with f(a_i)=b_i and f(R) contained in R.

1991 Mathematics Subject Classification: Primary 13B25, 13G05; Secondary 11B65, 41A05.

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