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

Abstract

AbstractLet D be a Krull domain admitting a prime element with finite residue field and let K be its quotient field. We show that for all positive integers k and $$1 < n_1 \le \cdots \le n_k$$ 1 < n 1 ≤ ⋯ ≤ n k , there exists an integer-valued polynomial on D, that is, an element of $${{\,\textrm{Int}\,}}(D) = \{ f \in K[X] \mid f(D) \subseteq D \}$$ Int ( D ) = { f ∈ K [ X ] ∣ f ( D ) ⊆ D } , which has precisely k essentially different factorizations into irreducible elements of $${{\,\textrm{Int}\,}}(D)$$ Int ( D ) whose lengths are exactly $$n_1, \ldots , n_k$$ n 1 , … , n k . Using this, we characterize lengths of factorizations when D is a unique factorization domain and therefore also in case D is a discrete valuation domain. This solves an open problem proposed by Cahen, Fontana, Frisch, and Glaz.