Towards a noncommutative Picard-Vessiot theory
Résumé
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.
Origine : Fichiers produits par l'(les) auteur(s)
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