Irreducibility of reciprocal polynomials and applications

Irreducibility of polynomials over the rationals is an important subject in mathematics, with many applications in computer algebra and algorithmics. This is a well established research area requiring different concepts and tools from many fields.

Although it is well known that “almost” every rational polynomial is irreducible, it is not easy to decide if a single polynomial is so. In this talk, we first recall some classical criteria for irreducibility. Then, we focus on the reciprocal case, and describe new irreducible criteria we have obtained in this context. This leads us to a new characterization of cyclotomic polynomials in terms of linear algebra. We finally discuss the possible consequences of our work in algorithmics and computer algebra.

Joint work with Eda Cesaratto (UNGS and CONICET).