{
  "abstract": "We prove Stieltjes' common root conjecture: Legendre polynomials of distinct degrees have no common nonzero root. We construct an auxiliary polynomial and show that, for the relevant power of two $q\\ge2$, the edge of slope $1/q$ in its $2$-adic Newton polygon has horizontal length less than $3q$. If a common nonzero root existed, Newton polygon theory and the Legendre differential equation would force the same edge to have horizontal length at least $3q$, yielding a contradiction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22330",
  "author_total": 1,
  "authors": [
    {
      "name": "Zikang Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0v6xanttg6p4t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22330",
  "fulltext_url": null,
  "id": "0v6xanttg6p4t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22330",
  "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.GM"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7214045636",
  "pdf_url": "https://arxiv.org/pdf/2609.22330",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0v6xanttg6p4t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22330",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22330"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "A Proof of the Common Root Conjecture for Legendre Polynomials",
  "title_text": "A Proof of the Common Root Conjecture for Legendre Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/0v6xanttg6p4t",
  "venue": "arXiv",
  "versions": []
}