{
  "abstract": "We study random polynomials of the form $R(x)=x^n+ω_{n-1}x^{n-1}+\\cdots+ω_0$, where $ω_0,\\dots,ω_{n-1}$ are independent, uniformly bounded integer-valued random variables, and $ω_1,\\dots,ω_{n-1}$ have a fixed common law $μ$. We prove (unconditionally) that, if the Rényi entropy of order $2$ satisfies $H_2(μ)=-\\log\\|μ\\|_2^2>12$, then $\\mathbb{P}(\\operatorname{disc}(R)\\text{ is a square})=O_μ(1/\\log n)$. Combined with previous results, this shows that, for such measures $μ$ and under additional hypotheses, $R$ has full Galois group $\\mathrm{Sym}(n)$ with high probability, when conditioned on $ω_0\\ne 0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40123",
  "author_total": 3,
  "authors": [
    {
      "name": "Guy Blachar"
    },
    {
      "name": "Emmanuel Breuillard"
    },
    {
      "name": "Gady Kozma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bncdsx0epxc7r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40123",
  "fulltext_url": null,
  "id": "bncdsx0epxc7r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40123",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7215120981",
  "pdf_url": "https://arxiv.org/pdf/2609.40123",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bncdsx0epxc7r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40123",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40123"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Galois groups of random polynomials of large degree",
  "title_text": "Galois groups of random polynomials of large degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/bncdsx0epxc7r",
  "venue": "arXiv",
  "versions": []
}