We prove that, for any Fuchsian system of differential equations on the Riemann sphere, there exists a rational matrix function whose partial indices coincide with the splitting type of the canonical vector bundle induced from the Fuchsian system. From this, we obtain solution of the Riemann-Hilbert boundary value problem for piecewise constant matrix function in terms of holomorphic sections of vector bundle and calculate the partial indices of the problem.

中文翻译：

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具有分段常数转移函数的Riemann-Hilbert边值问题

我们证明，对于Riemann球面上的任何Fuchsian微分方程组，都存在一个有理的矩阵函数，该函数的局部索引与从Fuchsian系统导出的规范向量束的分裂类型一致。由此，我们根据向量束的全纯部分，获得了分段常数矩阵函数的黎曼-希尔伯特边值问题的解，并计算了该问题的局部指数。