Title: The Galois group of a polynomial with not so large coefficients

Abstract: 

A random polynomial f with integer coefficients is expected to have a full Galois group, asymptotically almost surely (a.a.s.).
In the large box model, where the degree n=\deg f is fixed and the coefficients of f are chosen uniformly from the integers in the  box [-L,L]^n,  it goes back to Hilbert and to van-der Waerden the Galois group is full, a.a.s. In the restricted coefficients model, that is when n\to \infty  and L is fixed, it was only recently proven that f is irreducible a.a.s. (unconditionally if L\geq 17 and conditionally on GRH for L\geq 1). It is still open whether the Galois group is full a.a.s. The state of the art is that the Galois group is large in the sense that it is either the alternating group or the symmetric group. 

In this talk we will present a new result, joint with Noam Goldgraber, on a model that is as close as possible to the restricted coefficient model: f has a  full Galois group a.a.s., provided L=L(n)\to \infty, arbitrarily slow. We will discuss the main tools of the proof, and if time permits I will present some generalizations to general measures.

Zoom: https://huji.zoom.us/j/84202575300?pwd=QXBvNjV0bDBWUmwxVkFIYXpzQ29RQT09