Aspects Combinatoires des Polylogarithmes et des Sommes d'Euler-Zagier

Hoang Ngoc Minh

Séminaire lotharingien de combinatoire, Tome 43 (1999-2000)

Résumé

The algebra of polylogarithms is the smallest C-algebra which contains the constants and which is stable under integration with respect to the differential forms dz/z and dz/(1-z). It is known that this algebra is isomorphic to the algebra of the noncommutative polynomials equipped with the shuffle product. As a consequence, the polylogarithms Li_n(g(z)) with n>=1, where the g(z) belong to the group of biratios, are the polylogarithms indexed by Lyndon words with coefficients in a certain transcendental extension of Q: the algebra of the Euler-Zagier sums. We conjecture that this algebra is an algebra of polynomials, and we attempt to find a basis for this algebra. The question of knowing whether the polylogarithms Li_n(g(z)) satisfy a linear functional equation is effectively decidable up to a conjecture of Zagier about the dimension of the algebra. This decision procedure makes use of the decomposition of those polylogarithms indexed by the Lyndon basis. Such an algorithm is based on the factorisation of the generating function of these polylogarithms.