{
  "abstract": "Given a collection of algebraic numbers $\\mathcal{S}\\subset \\overline{\\mathbb{Q}}$ we study the varieties $V$ in $\\mathbb{A}^m_\\mathbb{C}$ such that $V(\\mathbb{C})\\cap\\mathcal{S}^m$ is Zariski-dense in $V$. We show that for many classical families of algebraic numbers $\\mathcal{S}$---such as the family of roots of generalized Laguerre polynomials $L_n^{(α)}(x)$, for a finite collection of $α\\in \\mathbb{Q}$---an unlikely intersections theorem holds. For example, in the case $m=2$, we prove that an irreducible curve in $\\mathbb{A}^2_\\mathbb{C}$ has infinitely many points from $\\mathcal{S}^2$ if and only if it is of the form $x_1=x_2,$ or $x_1=s$, or $x_2=s$ for a fixed $s \\in \\mathcal{S}$. This is an analogue of the classical theorems of Ihara, Serre, and Tate, treating the case of $\\mathcal{S}$ consisting of the roots of unity, and of the Manin--Mumford conjecture. We also show that a similar result holds almost surely for roots of a collection of random polynomials of growing degree and bounded height. The proofs rely on a uniform Galois-theoretic criterion ensuring the unlikely intersection property.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09354",
  "author_total": 2,
  "authors": [
    {
      "name": "Lior Bary-Soroker"
    },
    {
      "name": "Borys Kadets"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0a01qmgt6p15r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09354",
  "fulltext_url": null,
  "id": "0a01qmgt6p15r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09354",
  "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.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7212125432",
  "pdf_url": "https://arxiv.org/pdf/2609.09354",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0a01qmgt6p15r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09354",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09354"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generic Manin-Mumford",
  "title_text": "Generic Manin-Mumford",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/0a01qmgt6p15r",
  "venue": "arXiv",
  "versions": []
}