Skip to content
DraconiticResearch
Search Sources API
182,469 studies31 sourcesUpdated 20 min ago
PreprintCC BY 4.0arXiv

n-Valued Analogue of Abel's Theorem

  • Mikhail Kornev
Open at sourcePDFDOICite

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 ℙ1 with a strong two-sided unit is commutative. Over a reduced commutative ℚ-algebra, any one-dimensional formal n-valued group is commutative. We then classify n-valued formal groups over arbitrary 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 𝔾m. This proves Kholodov's classification conjecture over integral domain ℚ-algebras.

Related

  1. C0-Analogue of Ismagilov's Theorem
    Stéphane Tchuiaga
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  2. A Group Analogue of the Ghurye--Olkin--Ibragimov Theorem
    Gennadiy Feldman
    arXivPreprintarXiv licenceFull textProbabilityarXiv
  3. The Almost Stacked Hypothesis: A Conjectural Analogue of Sjöstrand's Cover Pebbling Theorem
    Tamás Csernák, Lajos Soukup
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  4. Self-Referential K-SAT and the Finite Analogue of Gödel's Incompleteness Theorem
    Wen Fang, Xianxian Li, Jun Liu +3
    arXivPreprintCC BY 4.0Full textGeneral mathematicsarXiv
  5. A Vector-Valued Extension of a Theorem of Marciszewski
    Leandro Candido
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{kornev2026nvalued,
  title = {{$n$-Valued Analogue of Abel's Theorem}},
  author = {Mikhail Kornev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14044},
  eprint = {2610.14044},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14044},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - n-Valued Analogue of Abel's Theorem
AU  - Mikhail Kornev
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.14044
UR  - https://arxiv.org/abs/2610.14044
AB  - 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 ℙ^1 with a strong two-sided unit is commutative. Over a reduced commutative ℚ-algebra, any one-dimensional formal n-valued group is commutative. We then classify n-valued formal groups over arbitrary 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 𝔾_m. This proves Kholodov's classification conjecture over integral domain ℚ-algebras.
ER  -