{
  "abstract": "We establish Berry-Esseen bounds for the number of real roots of Gaussian Weyl polynomials $P_n$, where $n$ denotes the degree and is assumed to be sufficiently large. For each fixed $B$ above an absolute threshold, let $I_n=[-\\sqrt n+B\\sqrt{\\log n}, \\sqrt n-B\\sqrt{\\log n}]$. Uniformly over deterministic compact intervals $I\\subseteq I_n$ whose length $\\ell$ is sufficiently large and depends on $n$, the distribution of the standardized number of real roots in $I$ has Kolmogorov distance at most $C\\log\\ell/\\sqrt\\ell$ from the standard Gaussian distribution. Consequently, every such interval sequence with $\\ell\\to\\infty$ satisfies a central limit theorem. In particular, taking $I=I_n$ gives the bound $C\\log n/n^{1/4}$. The same bound holds for the distribution of the standardized number of real roots on $\\mathbb R$. The key idea is to approximate the polynomial zero count by a sum of locally dependent random variables. We first couple the polynomial to a stationary Gaussian process and then truncate a moving-average representation of that process to obtain finite-range dependence. This strategy provides a route to Berry-Esseen bounds for other random polynomials with Gaussian coefficients whenever such a stationary approximation and quantitative truncation are available.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12734",
  "author_total": 3,
  "authors": [
    {
      "name": "Yuchen Wang"
    },
    {
      "name": "Dawei Lu"
    },
    {
      "name": "Song-Hao Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/25mx2d1zag5jf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12734",
  "fulltext_url": null,
  "id": "25mx2d1zag5jf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12734",
  "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"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7212856215",
  "pdf_url": "https://arxiv.org/pdf/2609.12734",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/25mx2d1zag5jf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12734",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12734"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Berry-Esseen Bounds for the Number of Real Zeros of Gaussian Weyl Polynomials",
  "title_text": "Berry-Esseen Bounds for the Number of Real Zeros of Gaussian Weyl Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/25mx2d1zag5jf",
  "venue": "arXiv",
  "versions": []
}