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

Abstract

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

arXiv admin note: text overlap with arXiv:2206.11003