Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements.

Let be a Krull domain admitting a prime element with finite residue field and let be its quotient field. We show that for all positive integers and , there exists an integer-valued polynomial on , that is, an element of , which has precisely essentially different factorizations into irreducible elements of whose lengths are exactly . Using this, we characterize lengths of factorizations when is a unique factorization domain and therefore also in case is a discrete valuation domain.

Integer-valued polynomials on valuation rings of global fields with prescribed lengths of factorizations.

Let be a valuation ring of a global field . We show that for all positive integers and there exists an integer-valued polynomial on , that is, an element of , which has precisely essentially different factorizations into irreducible elements of whose lengths are exactly . In fact, we show more, namely that the same result holds true for every discrete valuation domain with finite residue field such that the quotient field of admits a valuation ring independent of whose maximal ideal is principal or whose residue field is finite.