Towards a noncommutative Picard-Vessiot theory.

G. Duchamp (1) , Viincel Hoang Ngoc Minh , Vu Nguyen Dinh , Pierre Simonnet

1 LIPN - Laboratoire d'Informatique de Paris-Nord

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.

Mots clés: Picard-Vessiot theory; Differential equations; Rational series

Domaines: Calcul formel [cs.SC]; Combinatoire [math.CO]

G. Duchamp, Viincel Hoang Ngoc Minh, Vu Nguyen Dinh, Pierre Simonnet. Towards a noncommutative Picard-Vessiot theory.. 2021. ⟨hal-02921131v2⟩