{
  "abstract": "Ball's weak cylinder conjecture asserts that a set of $p^2$ points in $\\mathrm{AG}(3,p)$ with at least $p$ non-determined directions is a union of $p$ parallel lines. We prove the conjecture for sets admitting a polynomial graph representation of total degree at most $D$, provided that $p>D^{9/2}$. More precisely, a non-cylindrical graph of degree $D\\ge3$ satisfies $|\\mathcal{N}_S|\\le p+1-\\lceil p/(D-1)\\rceil$ under the same hypothesis. This bound is attained when $p\\equiv1\\pmod{D-1}$. The proof relates non-determined directions to permutation polynomials on parallel affine lines. A rigidity lemma for moving line restrictions, together with a reconstruction of their compositional components, bounds the number of contributing directions uniformly in $p$. Directions with affine restrictions are treated separately; all other contributions are bounded using geometric indecomposability and the Weil bound for plane curves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16803",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniele Bartoli"
    },
    {
      "name": "Alessandro Giannoni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5fhd5915nh8b4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16803",
  "fulltext_url": null,
  "id": "5fhd5915nh8b4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16803",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221780380",
  "pdf_url": "https://arxiv.org/pdf/2610.16803",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5fhd5915nh8b4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:16Z",
      "native_id": "2610.16803",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16803"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The weak cylinder conjecture for polynomial graphs of bounded degree",
  "title_text": "The weak cylinder conjecture for polynomial graphs of bounded degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:16Z",
  "url": "https://research.draconitic.com/paper/5fhd5915nh8b4",
  "venue": "arXiv",
  "versions": []
}