Sets of lengths of factorizations of integer-valued polynomials on De...

Non-unique factorization of elements into irreducibles has been observed in the ring of integer-valued polynomials and its generalizations.  We show that all finite (multi-)sets of natural numbers greater than 1 occur as (multi-)sets of lengths of a polynomial in $\mathrm{Int}(D)$ where $D$ is a Dedekind domain with infinitely many maximal ideals all of whose residue fields are finite. For given integers $1 < m_1 \le \ldots \le m_n$ we can explicitly construct an element $f\in \mathrm{Int}(D)$ which has exactly $n$ essentially different factorizations of lengths $m_1$,  \ldots, $m_n$.  In this talk, we speak about the construction techniques and consequences.

This is joint work with S.~Frisch and S.~Nakato.