{
  "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 $|a_j| \\leq B$ 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 $n \\geq 40$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22297",
  "author_total": 8,
  "authors": [
    {
      "name": "Theresa C. Anderson"
    },
    {
      "name": "Sam Chow"
    },
    {
      "name": "Rainer Dietmann"
    },
    {
      "name": "David Hokken"
    },
    {
      "name": "Peter Koymans"
    },
    {
      "name": "Robert J. Lemke Oliver"
    },
    {
      "name": "Will Sawin"
    },
    {
      "name": "Mark Shusterman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/db4fwz7ajxdje.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22297",
  "fulltext_url": null,
  "id": "db4fwz7ajxdje",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22297",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7204194257",
  "pdf_url": "https://arxiv.org/pdf/2608.22297",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/db4fwz7ajxdje.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2608.22297",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22297"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The strong form of Van der Waerden's conjecture via twisted Chowla",
  "title_text": "The strong form of Van der Waerden's conjecture via twisted Chowla",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/db4fwz7ajxdje",
  "venue": "arXiv",
  "versions": []
}