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 ≥ 3 and let En(B) be the number of monic, irreducible, non-Sn polynomials f = Xn + a1 Xn−1 + ⋯ + an with |aj| ≤ B for all j. A recent breakthrough of Bhargava bounds En(B) ≪ Bn−1. This spectacularly resolves a conjecture of Van der Waerden, but leaves open its stronger form, namely that En(B) = o(Bn−1). Inspired by recent progress, we now prove this stronger form for all n ≥ 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 An is o(Bn−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 ≥ 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 ≥ 40.
Related
BibTeX
@misc{anderson2026strong,
title = {{The strong form of Van der Waerden's conjecture via twisted Chowla}},
author = {Theresa C. Anderson and Sam Chow and Rainer Dietmann and David Hokken and Peter Koymans and Robert J. Lemke Oliver and Will Sawin and Mark Shusterman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22297},
eprint = {2608.22297},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22297},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The strong form of Van der Waerden's conjecture via twisted Chowla
AU - Theresa C. Anderson
AU - Sam Chow
AU - Rainer Dietmann
AU - David Hokken
AU - Peter Koymans
AU - Robert J. Lemke Oliver
AU - Will Sawin
AU - Mark Shusterman
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.22297
UR - https://arxiv.org/abs/2608.22297
AB - 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 ≥ 3 and let E_n(B) be the number of monic, irreducible, non-S_n polynomials f = X^n + a_1 X^n−1 + ⋯ + a_n with |a_j| ≤ B for all j. A recent breakthrough of Bhargava bounds E_n(B) ≪ 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 ≥ 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 ≥ 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 ≥ 40.
ER -