Irreducibility of random polynomials

Peter Varju
University of Cambridge

Let P be a random polynomial of degree d such
that the leading and constant coefficients are 1 and the rest
of the coefficients are independent random variables taking
the value 0 or 1 with equal probability.

Odlyzko and Poonen conjectured that P is irreducible with
probability tending to 1 as d grows. I will talk about an on-going joint
work with Emmanuel Breuillard, in which we prove that GRH implies
this conjecture.

The proof is based on estimates for the mixing time of
random walks on F_p, where the steps are given by the maps
x-> ax and x->ax+1 with equal probability.
The method may also apply to other families of polynomials.

Back to Workshop I: Expected Characteristic Polynomial Techniques and Applications