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

Let 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 \(1 < n_1 \le \cdots \le n_k\) $1<n_1\le \cdots \le n_k$ , there exists an integer-valued polynomial on D, that is, an element of \({{\,\textrm{Int}\,}}(D) = \{ f \in K[X] \mid f(D) \subseteq D \}\) $\text{Int}(D)=\{f\in K[X]\mid f(D)\subseteq D\}$ , which has precisely k essentially different factorizations into irreducible elements of \({{\,\textrm{Int}\,}}(D)\) $\text{Int}(D)$ whose lengths are exactly \(n_1, \ldots , n_k\) $n_1,\dots ,n_k$ . 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.