Irreducibility of the characteristic polynomial of a random integer matrix

Abstract: Consider a random polynomial with integer coefficients. A natural conjecture is that the polynomial is irreducible with high probability and its Galois group is S_n. This question has been studied for various models of random polynomial. The usual two models are the "bounded degree model", in which the degree is constant and the coefficients are large, and the "bounded height model", in which the coefficients are drawn uniformly from a fixed interval and the degree becomes large. We will study a variant of the bounded height model: take a large n x n matrix with independent +-1 entries and take its characteristic polynomial. To study this question we will combine ideas from the bounded height model with random matrix theory over a finite field. The method we use is dependent on both the extended Riemann hypothesis and the classification of finite simple groups.

number theory

Audience: researchers in the topic

CRM-CICMA Québec Vermont Seminar Series

Series comments: En ligne/Web - Pour information, veuillez communiquer à / For details, please contact: activités@crm.umontreal.ca