20260407 David Hokken

There are many tools to measure the ‘complexity’ of a polynomial P in Z[x]. One is the Galois group of P, which measures the symmetries of the zeros of the polynomial. Unfortunately, it is quite hard to calculate Galois groups, so it makes sense to randomize the setup and ask what the Galois group is of a 'typical’ polynomial. It is a long-held belief, going back in various forms to Hilbert, Van der Waerden, and Odlyzko—Poonen, that such typical P is irreducible and has a large Galois group over the rationals. We will discuss what this means exactly and highlight some recent progress in this area, in particular for reciprocal polynomials: those P that satisfy P(x) = x^n P(1/x), where n is the degree of P. Along the way, we encounter another complexity notion of polynomials — the Mahler measure — and discuss the special place that reciprocal polynomials have in the theory of Mahler measures.