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 Int(D)={f∈K[x]|f(D)⊆D} be the ring of integer-valued polynomials on D.Given any finite multiset {k1,…,kn} of integers greater than 1, we construct a polynomial in Int(D) which has exactly n essentially different factorizations into irreducibles in Int(D), the lengths of these factorizations being k1, …, kn. We also show that there is no transfer homomorphism from the multiplicative monoid of Int(D) to a block monoid.