Computing the Lie Algebra of the Differential Galois Group of a Linear Differential System

Moulay A. Barkatou (1) , Thomas Cluzeau (1) , Jacques-Arthur Weil (1) , Lucia Di Vizio (2)

1 XLIM-MATHIS - Mathématiques & Sécurité de l'information
2 LMV - Laboratoire de Mathématiques de Versailles

We consider a linear differential system [A] : y' = A y with coefficients in C(x). The differential Galois group G of [A] is a linear algebraic group which measures the algebraic relations among solutions. Although there exist general algorithms to compute G, none of them is either practical or implemented. This paper proposes an algorithm to compute the Lie algebra g of G when [A] is absolutely irreducible. The algorithm is implemented in Maple.

Mots clés: Computer algebra; Algorithms; Linear differential systems; Differential Galois theory; Lie algebras; Grothendieck-Katz p-curvature conjecture; Eigenrings; Reduced forms

Domaines: Calcul formel [cs.SC]

Moulay A. Barkatou, Thomas Cluzeau, Jacques-Arthur Weil, Lucia Di Vizio. Computing the Lie Algebra of the Differential Galois Group of a Linear Differential System. 41st ACM International Symposium on Symbolic and Algebraic Computation (ISSAC), Jul 2016, Waterloo, Canada. ⟨10.1145/2930889.2930932⟩. ⟨hal-01676487⟩