Squarefree values of polynomial discriminants
The question as to whether a positive proportion of monic irreducible integer polynomials of degree n have squarefree discriminant is an old one; an exact formula for the density was conjectured by Lenstra. (The interest in monic polynomials f with squarefree discriminant comes from the fact that in such cases Z[x]/(f(x)) gives the ring of integers in the number field Q[x]/(f(x)).)
In this talk, we will describe recent work with Arul Shankar and Xiaoheng Wang that allows us to determine the probability that a random monic integer polynomial has squarefree discriminant - thus proving the conjecture of Lenstra.