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

TL;DR

This paper constructs integer-valued polynomials over Dedekind domains with finite residue fields that have prescribed sets of factorization lengths and demonstrates the absence of transfer homomorphisms to block monoids.

Contribution

It introduces a method to explicitly construct polynomials with specified factorization length multisets in the ring of integer-valued polynomials on Dedekind domains.

Findings

Constructed polynomials with exactly n factorizations of prescribed lengths

Demonstrated the non-existence of transfer homomorphisms to block monoids

Provided insights into the factorization structure of integer-valued polynomial rings

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.