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

Abstract

Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\mathrm{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$ be the ring of integer-valued polynomials on $D$. Given any finite multiset $\{k_1,\dots ,k_n\}$ of integers greater than $1$, we construct a polynomial in $\mathrm{Int}(D)$ which has exactly $n$ essentially different factorizations into irreducibles in $\mathrm{Int}(D)$, 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 $\mathrm{Int}(D)$ to a block monoid.

References in corpus (2)

• Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics
• Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids

Cited by in corpus (12)

• Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics
• Factorization under Local Finiteness Conditions
• Semigroup rings as weakly Krull domains
• Absolute irreducibility of the binomial polynomials
• Non-absolutely irreducible elements in the ring of Integer-valued polynomials
• Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains
• Atomicity and Factorization of Puiseux Monoids
• A realization result for systems of sets of lengths
• A graph-theoretic criterion for absolute irreducibility of integer-valued polynomials with square-free denominator
• Factorization Theory in Commutative Monoids
• Primes and absolutely or non-absolutely irreducible elements in atomic domains
• Irreducible integer-valued polynomials with prescribed minimal power that factors non-uniquely