Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 2 min ago
PreprintarXiv licencearXiv

A Kac-Moody root system and linear ordinary differential equations

  • Toshio Oshima
Open at sourcePDFDOICite

Abstract

We show that every irreducible linear differential equations on the Riemann sphere with regular and/or unramified irregular singularities can be transformed into either the trivial equation or a Fuchsian system of E8-fundamental spectral type by a sequence of invertible transformations consisting of confluences, unfoldings, Laplace transformations, gauge transformations and Möbius transformations. The E8-spectral type is uniquely determined by the original equation. A Fuchsian system of E8-fundamental spectral type has three singular points, 0, 1, and ∞. The degrees of the minimal polynomials of the residue matrices at 0 and 1 are two and three, respectively, and the sum of the maximal dimensions of eigenspaces of the three residue matrices is not greater than the size of the matrices. The result follows from the correspondence between spectral types of Fuchsian systems and roots of a star-shaped Kac-Moody root system.

Related

  1. Rational linear forms of linearizable ordinary differential equations
    Dmitry Lyakhov
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryStatistical & nonlinear physicsarXiv
  2. Existence and multiplicity results for third-order ordinary differential equations with derivative dependence subject to the Riemann–Stieltjes integral boundary conditions
    Du Yue, Cui Yujun
    Demonstratio MathematicaPeer-reviewedCC BYFull textGeneral mathematicsDOAJ
  3. The specular ellipse method for scalar ordinary differential equations: exactness and accuracy up to fourth order
    Kiyuob Jung
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
  4. Cross-ratios and first order ordinary differential equations
    Oumar Wone
    arXivPreprintarXiv licenceFull textAnalysis & PDEsGeometry & topologyarXiv
  5. A Sensitivity-based Framework for Calibrating Coupled Ordinary Differential Equations under Model Discrepancy
    Mitchel J. Colebank, William Consagra
    arXivPreprintCC BY-NC-SA 4.0Full textOptimisation & controlStatisticsarXiv

Cite

BibTeXRIS
BibTeX
@misc{oshima2026kacmoody,
  title = {{A Kac-Moody root system and linear ordinary differential equations}},
  author = {Toshio Oshima},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39733},
  eprint = {2609.39733},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39733},
}
RIS
TY  - UNPB
TI  - A Kac-Moody root system and linear ordinary differential equations
AU  - Toshio Oshima
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39733
UR  - https://arxiv.org/abs/2609.39733
AB  - We show that every irreducible linear differential equations on the Riemann sphere with regular and/or unramified irregular singularities can be transformed into either the trivial equation or a Fuchsian system of E_8-fundamental spectral type by a sequence of invertible transformations consisting of confluences, unfoldings, Laplace transformations, gauge transformations and Möbius transformations. The E_8-spectral type is uniquely determined by the original equation. A Fuchsian system of E_8-fundamental spectral type has three singular points, 0, 1, and ∞. The degrees of the minimal polynomials of the residue matrices at 0 and 1 are two and three, respectively, and the sum of the maximal dimensions of eigenspaces of the three residue matrices is not greater than the size of the matrices. The result follows from the correspondence between spectral types of Fuchsian systems and roots of a star-shaped Kac-Moody root system.
ER  -