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Séminaire Lotharingien de Combinatoire, B43e (2000), 29 pp.

# Hoang Ngoc Minh

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

**Abstract.**
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.

Received: March 15, 1999; Accepted: August 31, 1999.

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