Session S09 - Number Theory in the Americas

Thursday, July 15, 11:00 ~ 12:00 UTC-3

Galois groups of random integer polynomials

Manjul Bhargava

Princeton , USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as can be seen by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known for $n\leq 4$ due to work of van der Waerden and Chow and Dietmann. In this talk, we describe how to prove van der Waerden's Conjecture for all degrees $n$.

View abstract PDF