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 = M S) 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.

In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0, x1} with coefficients in a Q-extension, A, subjected to some suitable conditions, there exists an unique algebra homomorphism ϕ from the Q-algebra generated by the convergent polyzêtas to A such that Φ is computed from ΦKZ Drinfel'd associator by applying ϕ to each coefficient. We prove ϕ exists and it is a free Lie exponential over X. Moreover, we give a complete description of the kernel of polyzêta and draw some consequences about a structure of the algebra of convergent polyzêtas and about the arithmetical nature of the Euler constant.

On the solutions of the universal differential equation with three regular singularities (On solutions of 3) Tome 11, n o 2 (2019), p. 25-64. © Les auteurs et Confluentes Mathematici, 2019. Certains droits réservés. Cet article est mis à disposition selon les termes de la licence Licence Internationale d'Attibution Creative Commons BY 4.0 https://creativecommons.org/licenses/by/4.0 L'accès aux articles de la revue « Confluentes Mathematici » (http://cml.centre-mersenne.org/) implique l'accord avec les conditions générales d'utilisation (http://cml.centre-mersenne. org/legal/). Confluentes Mathematici est membre du Centre Mersenne pour l'édition scientifique ouverte http://www.centre-mersenne.org/

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