AI 中文总结
本文证明黎曼球面上不可约线性微分方程可通过可逆变换化为平凡方程或$E_8$-基本谱型Fuchs系统，谱型由原方程唯一确定，并基于星形Kac-Moody根系建立对应关系。
AI 中文摘要
我们证明，在黎曼球面上具有正则和/或无分支不规则奇点的每个不可约线性微分方程，都可以通过由合流、展开、拉普拉斯变换、规范变换和莫比乌斯变换组成的可逆变换序列，转化为平凡方程或$E_8$-基本谱型的Fuchs系统。$E_8$-谱型由原方程唯一确定。$E_8$-基本谱型的Fuchs系统具有三个奇点，即0、1和$\infty$。在0和1处的留数矩阵的最小多项式的次数分别为2和3，并且三个留数矩阵的特征子空间的最大维数之和不大于矩阵的大小。该结果源于Fuchs系统的谱型与星形Kac-Moody根系的根之间的对应关系。
英文摘要
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 $\infty$. 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.
Comments44 pages
