Number Theory Seminar

There will be two talks in our NT Seminar at 10am and 11am, respectively:

Speaker: Alexandre Roy 

Title: The number of number fields with fixed degree and bounded discriminant

Abstract: A folk conjecture is that the number of number fields with fixed degree and discriminant smaller than X in absolute value grows as a constant times X. I will present proofs from Bhargava of this fact on some specific degree and I will present some lower and upper bounds on the general case from Ellenberg and Venkatesh, Couveignes, Lemke and Thorne. In particular, I will use results from Minkowski on lattices and show how it can be used in this situation.


Speaker: Leo Dubocs

Title:  A proof of the Schinzel-Zassenhaus conjecture, following V. Dimitrov

Abstract: In a recent breakthrough, Vesselin Dimitrov proved the following result: If P is an integer, monic, irreducible, non-cyclotomic polynomial of degree n, then its roots \alpha_i satisfy : max|\alpha_i| \geq 2^{1/4n}. This proves a conjecture due to A. Schinzel and H. Zassenhaus, dating back to the 60s. In this talk, I will explain the tools used in Dimitrov's approach, in particular the notion of transfinite diameter, then I will give his proof of the conjecture.