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

TL;DR

This paper investigates the lengths of factorizations of integer-valued polynomials over Krull domains with prime elements, providing explicit constructions and characterizations that solve an open problem in the field.

Contribution

It constructs integer-valued polynomials with prescribed factorization lengths and characterizes these lengths in UFDs and DVRs, addressing an open problem.

Findings

Existence of polynomials with exactly k factorizations of specified lengths

Characterization of factorization lengths in UFDs and DVRs

Solution to an open problem in factorization theory

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 $1 < n_1 \leq \ldots \leq n_k$ 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 $n_1,\ldots,n_k$. 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.