{
  "abstract": "Combining local boundary analysis, algebraic curves and the computational statements specified in the appendix, we prove that the planar Newtonian five-body problem has finitely many central configurations up to rotation, translation and scaling for every choice of positive masses. This gives the five-body statement of Smale's sixth problem. We use the singular-sequence classification of Albouy and Kaloshin and resolve the corresponding boundary equations through weighted blow-up charts, rank calculations and local compatibility conditions. For the two remaining diagrams, we prove that the product $Δ$ of all squared mutual distances vanishes. On a hypothetical solution curve containing a positive arc, a pole of a four-product therefore gives a zero of $Δ$. A pole of $Δ$ elsewhere on the completed curve, together with $Δ^4=\\prod_{i<j}p_{ij}$, then contradicts the same local vanishing result. The computational statements for Diagrams~2, 3 and~6 involve exact algebra, interval arithmetic and ball arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16107",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhe Chen"
    },
    {
      "name": "Zhicheng Tong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/929aj010b8p7v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16107",
  "fulltext_url": null,
  "id": "929aj010b8p7v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16107",
  "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-ph"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7221772022",
  "pdf_url": "https://arxiv.org/pdf/2610.16107",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/929aj010b8p7v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:22Z",
      "native_id": "2610.16107",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16107"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Finiteness of planar central configurations of five bodies for all positive masses: On Smale's sixth problem",
  "title_text": "Finiteness of planar central configurations of five bodies for all positive masses: On Smale's sixth problem",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/929aj010b8p7v",
  "venue": "arXiv",
  "versions": []
}