{
  "abstract": "Let $\\Fq$ be a finite field, let $F,G\\in\\Fq[T]$ be nonconstant, and let $J\\in\\Fq[T]$. We study the parameterized two-parameter family \\[ C_{a,b}=\\{(x,t)\\in\\Fq^2:t=J(x)+aF(x)+bG(x)+λab\\}, \\qquad (a,b)\\in\\Fq^2, \\] where $λ\\in\\Fq^\\times$, with incidences counted with parameter multiplicity. If $d=°F$, $e=°G$, and $\\charac(\\Fq)>d+e$, we prove the uniform spectral estimate \\[ \\left|I(\\cP,Ω)-\\frac{|\\cP||Ω|}{q}\\right| \\ll_{d,e}q^{5/8}\\sqrt{|\\cP||Ω|} \\] for arbitrary $\\cP,Ω\\subseteq\\Fq^2$. The proof combines two additive Fourier transforms with a fourth-moment estimate for Kloosterman matrices. If $1\\le d,e\\le2$ and $\\charac(\\Fq)>3$, midpoint--difference coordinates and finite Weyl quantization sharpen the error to the Vinh scale $O(q^{1/2}\\sqrt{|\\cP||Ω|})$. We further show that the latter phenomenon is intrinsic to the coefficient curve rather than to the degree of a chosen parametrization. There is a polynomial Lüroth factorization \\[ F=f\\circ H,\\qquad G=g\\circ H,\\qquad \\Fq(f,g)=\\Fq(T), \\] unique up to an affine change of $H$. If $Γ$ is the plane curve parametrized properly by $(f,g)$ and \\[ κ_H=\\max_{s\\in\\Fq}|H^{-1}(s)|, \\] then finite-fibre pullback multiplies the centered incidence norm by at most $\\sqrt{κ_H}$. Consequently, if $\\degΓ\\le2$ and $\\charac(\\Fq)>3$, then \\[ \\left|I(\\cP,Ω)-\\frac{|\\cP||Ω|}{q}\\right| \\ll \\sqrt{κ_H}\\,q^{1/2}\\sqrt{|\\cP||Ω|}, \\] even when $F$ and $G$ have arbitrarily large raw degree. The resulting expansion estimates for $J(x)+yF(x)+zG(x)+λyz$ hold without lower bounds on the individual input sets. In the balanced regime, the $q^3$ missing-value scale for this family is already covered by a theorem of Arala--Chow; the incidence theorems here additionally yield expansion consequences in strongly unbalanced regimes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03819",
  "author_total": 1,
  "authors": [
    {
      "name": "Yao Zhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d58p79p4rc1ya.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03819",
  "fulltext_url": null,
  "id": "d58p79p4rc1ya",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03819",
  "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"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7220648132",
  "pdf_url": "https://arxiv.org/pdf/2610.03819",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d58p79p4rc1ya.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03819",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03819"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Spectral Incidence Bounds over Finite Fields",
  "title_text": "Spectral Incidence Bounds over Finite Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/d58p79p4rc1ya",
  "venue": "arXiv",
  "versions": []
}