Title: The characteristic polynomial of a random matrix
Speaker: Sean Eberhard - Cambridge University
Abstract: Consider an n x n matrix M with independent random +-1 entries. The probability that M is singular is well-studied, and is known to be exponentially small (Kahn--Komlos--Szemeredi). In this talk we make the stronger claim that the characteristic polynomial phi of M is irreducible, and that the Galois group Gal(phi) is at least the alternating group, with high probability. The proof depends on both the extended Riemann hypothesis and the classification of finite simple groups (but if the entries are instead drawn uniformly from {1, ..., 210} then we can drop both dependencies). The method is related to recent work of Bary-Soroker--Kozma and Breuillard--Varju in the context of random polynomials, and to work of Maples and Nguyen--Paquette on random matrices mod p.