{
  "abstract": "Abel's classical theorem shows that a continuously differentiable one-dimensional associative law with a two-sided unit is locally equivalent to addition under a change of coordinate. We prove an $n$-valued analogue in the algebraic and formal settings. Over a field of characteristic zero, every associative algebraic $n$-valued monoid on $\\mathbb P^1$ with a strong two-sided unit is commutative. Over a reduced commutative $\\mathbb Q$-algebra, any one-dimensional formal $n$-valued group is commutative. We then classify $n$-valued formal groups over arbitrary $\\mathbb Q$-algebras that are integral domains: for any $n$-valued formal group there is a unique divisor $m$ of $n$ such that the law is strongly isomorphic to the $n/m$-fold diagonal of the cyclic elementary $m$-valued law $\\mathbb G_m$. This proves Kholodov's classification conjecture over integral domain $\\mathbb Q$-algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14044",
  "author_total": 1,
  "authors": [
    {
      "name": "Mikhail Kornev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fkysch1w4f4cd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14044",
  "fulltext_url": null,
  "id": "fkysch1w4f4cd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14044",
  "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.GR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221791795",
  "pdf_url": "https://arxiv.org/pdf/2610.14044",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fkysch1w4f4cd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14044",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14044"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$n$-Valued Analogue of Abel's Theorem",
  "title_text": "n-Valued Analogue of Abel's Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/fkysch1w4f4cd",
  "venue": "arXiv",
  "versions": []
}