关于Levelt定理

摘要：

设$(E)$是$\mathbb{P}^{1}(\mathbb{C}) $上的一个阶为$n$的齐次线性微分方程Fuchsian。Riemann（1857）的思想是通过研究局部系统来获得（$E$）解的性质。因此，他通过研究$\mathbb{P}^{1}(\mathbb{C}) \backslash\{3 points\} $上相关的秩为2的局部系统，得到了高斯超几何函数的一些性质。例如，他获得了库默尔变换的超几何函数没有任何计算。Riemann方法的成功是由于$\mathbb{P}^{1}(\mathbb{C}) \backslash\{3 points\} $上的不可约秩为2的局部系统在Katz \cite{Katz}意义下是线性刚性的。这一结果构成了线性刚性系统的最佳研究实例之一，它由Levelt定理\cite{B}定理1.2.3证明。在这项工作中，我们提出了一个部分推广的Levelt定理。

英文摘要：

Let $(E)$ be a homogeneous linear differential equation Fuchsian of order $n$ over $\mathbb{P}^{1}(\mathbb{C}) $. The idea of Riemann (1857) was to obtain the properties of solutions of ($E$) by studying the local system. Thus, he obtained some properties of Gauss hypergeometric functions by studying the associated rank 2 local system over $\mathbb{P}^{1}(\mathbb{C}) \backslash\{3 points\} $. For example, he obtained the Kummer transformations of the hypergeometric functions without any calculation. The success of the Riemann's methods is due to the fact that the irreducible rank 2 local system over $\mathbb{P}^{1}(\mathbb{C}) \backslash\{3 points\} $ is linearly "rigid" in the sense of Katz \cite{Katz}. This result constitute one of the best studied example of linear rigid system, it was proved by the Levelt's theorem \cite{B} Theorem 1.2.3. In this work we propose a partial generalization of the Levelt's theorem.

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