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. This solves an open problem proposed by Cahen, Fontana, Frisch, and Glaz.