Thursday, May 19, 2022 4:30 PM
Manjul Bhargava (Princeton)

Abstract: Of the (2H+1)^n monic integer polynomials

  f(x) = x^n + a_1 x^{n-1}+ ... + a_n

with max{|a_1|,...,|a_n|} ≤ H, how many have Galois group that is not the full symmetric group S_n?

There are clearly >> H^{n-1} such polynomials, as may be obtained by setting a_n = 0. In 1936, van der Waerden conjectured that H^{n-1} should in fact be the correct order of magnitude for the count of such polynomials.

The conjecture has been known previously for degrees n ≤ 4, due to work of van der Waerden and Chow and Dietmann.  In this talk, we will describe a proof of van der Waerden's Conjecture for all degrees n.