{
  "abstract": "We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\\in\\mathrm{Mat}_n(\\mathbb{F}_p)$ has independent and $α_n$-balanced entries. We prove that if $nα_n-\\log n\\to+\\infty$, then, for every fixed collection of distinct monic irreducible polynomials over $\\mathbb{F}_p$, the corresponding primary partitions of $A_n$ converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform random matrix model. This strengthens the finite-field cokernel universality theorem of Lee (arXiv:2511.13070), which assumes $α_n=c\\log n/n$ for a fixed $c>1$, by resolving the additive endpoint case explicitly left open in his work. Moreover, our result extends universality from ordinary cokernels to the full rational canonical form. The condition is optimal for universality uniformly over fixed irreducible polynomials. Our proof uses the surjection moment method developed by Sawin and Wood (arXiv:2210.06279) over $\\mathbb{F}_p[t]$, together with a refinement of Lee's Fourier estimates that retains the full leading-order decay. We also construct degree-$d$ critical sparse obstructions, suggesting a polynomial-dependent threshold of $1/d$ for irreducible polynomials of minimal degree $d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01413",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiahe Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dzfnbbevnve8f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01413",
  "fulltext_url": null,
  "id": "dzfnbbevnve8f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01413",
  "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.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7206212596",
  "pdf_url": "https://arxiv.org/pdf/2609.01413",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dzfnbbevnve8f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.01413",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01413"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp Threshold for Universality of Rational Canonical Forms over a Finite Field",
  "title_text": "Sharp Threshold for Universality of Rational Canonical Forms over a Finite Field",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/dzfnbbevnve8f",
  "venue": "arXiv",
  "versions": []
}