A Chen generating series, along a path and with respect to $m$ differential forms,is a noncommutative series on $m$ letters and with coefficients which are holomorphic functionsover a simply connected manifold in other words a series with variable (holomorphic) coefficients.Such a series satisfies a first order noncommutative differential equation which is considered, bysome authors, as the universal differential equation, (i.e.) universality can beseen by replacing each letter by constant matrices (resp. analytic vector fields)and then solving a system of linear (resp. nonlinear) differential equations.Via rational series, on noncommutative indeterminates and with coefficients in rings, andtheir non-trivial combinatorial Hopf algebras, we give the first step of a noncommutativePicard-Vessiot theory and we illustrate it with the case of linear differential equationswith singular regular singularities thanks to the universal equation previously mentioned.

中文翻译：

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走向非交换的 Picard-Vessiot 理论

沿着路径并关于 $m$ 微分形式的 Chen 生成级数是 $m$ 字母上的非交换级数，其系数是单连接流形上的全纯函数，换句话说，是具有可变（全纯）系数的级数。这样的级数满足一阶非对易微分方程，一些作者将其视为通用微分方程，（即）通过将每个字母替换为常数矩阵（分别为解析向量场）然后求解线性方程组可以看出普遍性(resp。非线性)微分方程。通过有理级数，非交换不定数和环中的系数，以及它们的非平凡组合Hopf代数，我们给出了非对易 Picard-Vessiot 理论的第一步，并且由于前面提到的通用方程，我们通过具有奇异正则奇点的线性微分方程的情况来说明它。