Towards a noncommutative Picard-Vessiot theory

TL;DR

This paper initiates a noncommutative Picard-Vessiot theory using rational series and combinatorial Hopf algebras, extending classical differential equations to a noncommutative setting with applications to linear equations with regular singularities.

Contribution

It introduces the first steps towards a noncommutative Picard-Vessiot theory using rational series and Hopf algebras, and illustrates it with linear differential equations with singularities.

Findings

Develops a framework for noncommutative differential equations using rational series.

Connects noncommutative series to classical linear differential equations with singularities.

Provides a foundational step towards a noncommutative Galois theory for differential equations.

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 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.