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arXiv · 2308.14535

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

Victor Fadinger·Daniel Windisch
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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.

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Victor Fadinger, Daniel Windisch. 2023-08-28. Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements. https://arxiv.org/abs/2308.14535

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