Independence of hyperlogarithms over function fields via algebraic combinatorics.

Matthieu Deneufchâtel (1) , Gérard Henry Edmond Duchamp (1) , Vincel Hoang Ngoc Minh (1) , Allan I. Solomon (2, 3)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 Department of Physics and Astronomy [Milton Keynes]
3 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: Noncommutative differential equations; Hyperlogarithms; Fuchsian-type equations.; Fuchsian-type equations

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

Matthieu Deneufchâtel, Gérard Henry Edmond Duchamp, Vincel Hoang Ngoc Minh, Allan I. Solomon. Independence of hyperlogarithms over function fields via algebraic combinatorics.. 2011. ⟨hal-00558773v1⟩