Independence of hyperlogarithms over function fields via algebraic combinatorics.

Matthieu Deneufchâtel (1) , Gérard H.E. Duchamp (2) , Vincel Hoang Ngoc Minh (1) , Allan I. Solomon (3, 4)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 LIPN - Laboratoire d'Informatique de Paris-Nord
3 Department of Physics and Astronomy [Milton Keynes]
4 LPTMC - Laboratoire de Physique Théorique de la Matière Condensée

We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor $M$ in the differential equation $dS=MS$) has only singularities of first order (Fuchsian-type equations) and this implies that they freely span a space which contains no primitive. We give direct applications where we extend the property of linear independence to the largest known ring of coefficients.

Mots clés: Fuchsian-type equations; Noncommutative differential equations; Hyperlogarithms; Fuchsian-type equations.

Domaines: Combinatoire [math.CO]; Calcul formel [cs.SC]; Physique mathématique [math-ph]

Matthieu Deneufchâtel, Gérard H.E. Duchamp, Vincel Hoang Ngoc Minh, Allan I. Solomon. Independence of hyperlogarithms over function fields via algebraic combinatorics.. Lecture Notes in Computer Science, Volume 6742 (2011),, 2011. ⟨hal-00558773v4⟩