{
  "abstract": "For a random polynomial, the number of real zeros $N_{\\mathbb R}$ is a highly nonlinear function of its coefficients, and its statistical properties have been studied extensively. A natural question is whether $N_{\\mathbb R}$ satisfies a central limit theorem. For various ensembles with iid standard Gaussian coefficients, such central limit theorems have been established; see, for instance, arXiv:1911.12182, arXiv:1801.06331, arXiv:1401.5745; Azais and Leon, Electron. J. Probab. 18 (2013), no. 68; arXiv:1504.05355, arXiv:2111.09015, arXiv:1707.09276, arXiv:1005.4113. These results rely on a rich range of tools, including Kac-Rice formulas, moment methods, and Wiener chaos decompositions. In the non-Gaussian setting, however, many of these tools are unavailable. To the best of our knowledge, prior central limit theorems beyond the Gaussian setting were limited to Kac-type polynomials, including hyperbolic polynomials; see the works of Maslova (1974), O. Nguyen and Vu (arXiv:1904.04347), and, more recently, Do, N. Nguyen, and O'Rourke (arXiv:2605.26402). In this paper, we prove a central limit theorem for the total number of real zeros of Weyl polynomials whose coefficients are iid copies of a symmetric, mean-zero, variance-one subgaussian random variable $ξ$. This substantially extends one of the main results of Do and Vu (arXiv:1707.09276) to a broad class of non-Gaussian distributions, including the Rademacher distribution. Without the symmetry assumption, we prove central limit theorems for the number of real zeros for positive bulk intervals, as well as for $[0,\\infty)$. Our proof combines the uniform one-point anti-concentration estimates from our recent work (arXiv:2511.07735) with the localization of Weyl polynomials around the coefficient index $i\\approx x^2$. While our proofs use comparison to compute the variances, the CLT deduction is rather direct.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06754",
  "author_total": 2,
  "authors": [
    {
      "name": "Ander Aguirre"
    },
    {
      "name": "Hoi H. Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9jz1pdv73azz4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06754",
  "fulltext_url": null,
  "id": "9jz1pdv73azz4",
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  "landing_url": "https://arxiv.org/abs/2609.06754",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
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    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7212027317",
  "pdf_url": "https://arxiv.org/pdf/2609.06754",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9jz1pdv73azz4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06754",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06754"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Central limit theorem for real zeros of random Weyl polynomials with general coefficients",
  "title_text": "Central limit theorem for real zeros of random Weyl polynomials with general coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/9jz1pdv73azz4",
  "venue": "arXiv",
  "versions": []
}