{
  "abstract": "We study the Galois group $G_f$ of a random polynomial $f$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x) = x^{2n}/b^n \\cdot f(b/x)$. We use a Euclidean height adapted to this relation. Our main result is an asymptotic theorem of van der Waerden--Bhargava type: for fixed $b \\neq 0$ and $n \\geq 4$, the number of polynomials of height at most $H$ whose Galois group is not the full hyperoctahedral group $S_2 \\wr S_n$ is an explicit constant times $H^n\\log H$, with an error of order $H^n$. We determine the dependence of the leading constant on $b$; the leading-order group $G_1$ is of index $2$. This paper is a sequel to a recent paper by Anderson, Bertelli, and the author addressing reciprocal polynomials (i.e. the case $b = 1$).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31613",
  "author_total": 1,
  "authors": [
    {
      "name": "Evan M. O'Dorney"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/093466qs0kvq0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31613",
  "fulltext_url": null,
  "id": "093466qs0kvq0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31613",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214633812",
  "pdf_url": "https://arxiv.org/pdf/2609.31613",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/093466qs0kvq0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31613",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31613"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Galois groups of twisted reciprocal polynomials: a uniform asymptotic",
  "title_text": "Galois groups of twisted reciprocal polynomials: a uniform asymptotic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/093466qs0kvq0",
  "venue": "arXiv",
  "versions": []
}