{
  "abstract": "We prove a quantitative polynomial Roth theorem for corners in \\(\\mathbb F_p^2\\) for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer $d$, there are constants $p_0$ and $C$ (depending only on $p$) so that for every $ p>p_0$, if polynomials \\(ϕ_1,ϕ_2\\in \\mathbb \\mathbb{F}_p [y]\\) are of degree $\\leq d$ vanishing at $0$ and are not linearly dependent, then every \\(A\\subset\\mathbb F_p^2\\) with $ |A|\\ge C p^{2-1/14} $ contains a nontrivial corner $$ (x_1,x_2),\\qquad (x_1+ϕ_1(y),x_2),\\qquad (x_1,x_2+ϕ_2(y)) $$ for some \\(y\\in\\mathbb F_p^\\times\\). This improves the estimate $p^{2-1/16}$ of Han--Lacey--Yang and removes the distinct-degree restriction from their quantitative theorem. The main obstruction is the equal-degree resonant case, where the Jacobian argument of Han--Lacey--Yang degenerates. We adjoin the frequency-independent part of the phase to form an augmented map \\(\\widetilde F:W\\to\\mathbb A^3\\) from the correlation threefold. We prove that this map is generically finite on every top-dimensional geometric component and has no two-dimensional fibre. Using the associated Artin--Schreier sheaf and Katz--Laumon estimates for Fourier transform of perverse sheaves, we obtain square-root cancellation outside an algebraic exceptional set of dimension at most one and uniformly bounded degree. A separate curve-sum argument gives uniform control on the exceptional set. An \\(\\ell^2\\) matrix estimate adapted to such sets completes the resonant case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13891",
  "author_total": 4,
  "authors": [
    {
      "name": "Ji Li"
    },
    {
      "name": "Chun-Yen Shen"
    },
    {
      "name": "Tuyen Trung Truong"
    },
    {
      "name": "Liangchuan Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dpa51tccz3q2d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13891",
  "fulltext_url": null,
  "id": "dpa51tccz3q2d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13891",
  "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.AG"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213326037",
  "pdf_url": "https://arxiv.org/pdf/2609.13891",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dpa51tccz3q2d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13891",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13891"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Polynomial corners in finite fields beyond the distinct-degree case",
  "title_text": "Polynomial corners in finite fields beyond the distinct-degree case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/dpa51tccz3q2d",
  "venue": "arXiv",
  "versions": []
}