Sets of lengths of factorizations of integervalued 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 $\operatorname{Int}(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integervalued 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 arXiv:
 arXiv:1710.06783
 Bibcode:
 2017arXiv171006783F
 Keywords:

 Mathematics  Commutative Algebra;
 13A05;
 13B25;
 13F20;
 11R04;
 11C08
 EPrint:
 doi:10.1016/j.jalgebra.2019.02.040