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Zbl 0959.68144
Hoang Ngoc Minh; Petitot, Michel
(Hoang Ngoc Minh, V.)
Lyndon words, polylogarithms and the Riemann $\zeta$ function. (English)
[J] Discrete Math. 217, No.1-3, 273-292 (2000). ISSN 0012-365X

Summary: The algebra of polylogarithms (iterated integrals over two differential forms $\omega_0= dz/z$ and $\omega_1= dz/(1- z))$ is isomorphic to the shuffle algebra of polynomials on non-commutative variables $x_0$ and $x_1$. The Multiple Zeta Values (MZVs) are obtained by evaluating the polylogarithms at $z= 1$. From a second shuffle product, we compute a Gröbner basis of the kernel of this evaluation morphism. The completeness of this Gröbner basis up to order 12 is equivalent to the classical conjecture about MZVs. We also show that certain known relations on MZVs hold for polylogarithms.

MSC 2000:: *11M32
11G55 Polylogarithms and relations with K-theory
68R15 Combinatorics on words

Keywords: multiple zeta values

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