Matrix-rank approaches to factorization problems in rings of integer-...

Non-unique factorization of elements into irreducibles has been observed in the ring of integer-valued polynomials and its generalizations. A particularity of non-UFDs is that there is 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--a property in between irreducibilty and primality. 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.

We particularly focus on the absolute irreducibility of the binomial polynomials in $Int(Z)$.