{
  "abstract": "We study the distribution of the zeros of $\\det P_N(z)$ where $P_N(z)$ is a random monic polynomial matrix, i.e., $P_N(z)=z^dI-\\sum_{j=0}^{d-1}A_{j,N}z^j$ for possibly coupled random matrices $A_{j,N}$, scaled to have entrywise variance $O(1/N)$. We provide general conditions under which this distribution almost-surely weakly converges as $N\\to\\infty$ to a deterministic measure, depending on just the variance and covariance of the entries of the $A_j$. This generalizes the circular, elliptic, and semicircle laws, which concern the special case of this question where $d=1$. Unlike those classical laws, these measures can combine nonuniform two-dimensional densities with singular components supported on curves, producing a variety of petal-shaped regions, inspiring our name ``the petaloid law''. We give explicit formulas for the limiting densities and supports. Under a Gaussianity assumption, we also show that there are almost surely no eigenvalues outside small neighborhoods of the limiting support.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40232",
  "author_total": 2,
  "authors": [
    {
      "name": "Rikhav Shah"
    },
    {
      "name": "Edward Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dtf6jzdzsv6ad.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40232",
  "fulltext_url": null,
  "id": "dtf6jzdzsv6ad",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40232",
  "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.PR"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7215101838",
  "pdf_url": "https://arxiv.org/pdf/2609.40232",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dtf6jzdzsv6ad.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.40232",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40232"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Spectra of Random Polynomial Matrices: the Petaloid Law",
  "title_text": "Spectra of Random Polynomial Matrices: the Petaloid Law",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/dtf6jzdzsv6ad",
  "venue": "arXiv",
  "versions": []
}