# Towards a noncommutative Picard-Vessiot theory

Abstract : 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 functions over 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, by some authors, as the universal differential equation, \textit{i.e.} universality can be seen 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, and their non-trivial combinatorial Hopf algebras, we give the first step of a noncommutative Picard-Vessiot theory and we illustrate it with the case of linear differential equations with singular regular singularities thanks to the universal equation previously mentioned.
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Cited literature [51 references]

https://hal.archives-ouvertes.fr/hal-02921131
Contributor : Gérard Duchamp <>
Submitted on : Monday, August 24, 2020 - 8:09:23 PM
Last modification on : Thursday, August 27, 2020 - 3:22:40 AM
Long-term archiving on: : Tuesday, December 1, 2020 - 6:32:53 AM

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Picard_Vessiot_NC_v9.pdf
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### Identifiers

• HAL Id : hal-02921131, version 1
• ARXIV : 2008.10872

### Citation

G. Duchamp, Viincel Hoang Ngoc Minh, Vu Nguyen Dinh. Towards a noncommutative Picard-Vessiot theory. 2020. ⟨hal-02921131⟩

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