Powers of irreducibles in rings of integer-valued polynomials

Non-unique factorization of elements into irreducibles has been observed in the ring of integer-valued polynomials and its generalizations. It is known that every (multi-)set consisting of integers strictly greater than 1 can be realized as the (multi-)set of lengths of an integer-valued polynomial over a Dedekind domain D with infinitely many maximal ideals whose residue fields are finite. Moreover, under the same assumptions on D, Int(D) is not transfer Krull. The proofs of these two statements are build on the constructions of integer-valued polynomials whose factorization behavior can be fully controlled. For this, it is crucial to avoid the situation of a factorization in which an irreducible factor occurs more than once. This is because in non-unique factorization domains there is in general no saying how the powers of an irreducible element factor. From a factorization-theoretic point of view, one therefore wants to identify those elements among the irreducibles whose powers factor uniquely. We call such elements absolutely irreducible. This talk provides an overview on factorization-theoretic aspects in rings of integer-valued polynomials with focus on absolutely irreducible elements, including recent results from joint work with Sophie Frisch, Moritz Hiebler, Sarah Nakato, and Daniel Windisch.