Joint IAS/Princeton University Number Theory Seminar

The problem of the density of squarefree discriminant polynomials is an old one, being considered by many people, and the density being conjectured by Lenstra. A proof has been out of question for a long time. The reason it was desired is that a squarefree discriminant polynomial $f$ immediately gives the ring of integers of $\mathbb Q[x]/f(x)$ and its Galois group. In recent joint work with Manjul Bhargava and Arul Shankar, we counted the number of odd degree polynomials with squarefree discriminant and proved the conjecture of Lenstra. In this talk, I will explain the general strategy of the squarefree sieve and the specific strategy to deal with discriminants which in turn leads to counting integral orbits for a representation of a non-reductive group.