The strong form of Van der Waerden's conjecture via twisted Chowla

Abstract

Determining the properties of a random polynomial has fuelled significant investigation over the past century. One driving force of this research is a 1936 paper of Van der Waerden. Fix $n\ge 3$ and let $E_n(B)$ be the number of monic, irreducible, non-$S_n$ polynomials $f=X^n+a_1{X}^{n-1}+\cdots +a_n$ with $|a_j|\le B$ for all $j$. A recent breakthrough of Bhargava bounds $E_n(B)\ll B^{n-1}$. This spectacularly resolves a conjecture of Van der Waerden, but leaves open its stronger form, namely that $E_n(B)=o(B^{n-1})$. Inspired by recent progress, we now prove this stronger form for all $n\ge 40$. Bhargava's result, together with work of Chow and Dietmann, essentially reduces the strong Van der Waerden conjecture to the claim that the number of polynomials $f$ with Galois group $A_n$ is $o(B^{n-1})$. We connect this claim to Chowla's conjecture, an area driving major mathematical progress. Indeed, we formulate a twisted function field version of Chowla's conjecture, and show how Van der Waerden's conjecture reduces to this conjecture for $n\ge 8$. Finally, we make significant progress towards this twisted Chowla conjecture, leading to our resolution of the strong form of Van der Waerden's conjecture for all $n\ge 40$.

v2: 25 pages; two authors added; proof of Van der Waerden's conjecture is now unconditional for $n\ge 40$