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arXiv · 1710.06783

Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields

Sophie Frisch·Sarah Nakato·Roswitha Rissner
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Abstract

Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\operatorname{Int}(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integer-valued polynomials on $D$. Given any finite multiset $\{k_1, \ldots, k_n\}$ of integers greater than $1$, we construct a polynomial in $\operatorname{Int}(D)$ which has exactly $n$ essentially different factorizations into irreducibles in $\operatorname{Int}(D)$, the lengths of these factorizations being $k_1$, \ldots, $k_n$. We also show that there is no transfer homomorphism from the multiplicative monoid of $\operatorname{Int}(D)$ to a block monoid.

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Sophie Frisch, Sarah Nakato, Roswitha Rissner. 2019-03-28. Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields. https://doi.org/10.1016/j.jalgebra.2019.02.040

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