Title: "The algorithmic and combinatoric aspects of functional equations for polylogarithms."

Abstract: The algebra of polylogarithms is the smallest $\bf C$-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 $n\ge1$, where the $g(z)$ belong to the group of biratios, are polynomial on the polylogarithms indexed by Lyndon words with coefficients in a certain transcendental extension of $\bf Q$~: $\bf MZV$, the algebra of the Euler-Zagier sums. The question of knowing whether the polylogarithms $Li_n(g(z))$ satisfy a linear functional equation is effectively decidable up to a construction of a basis for the algebra $\bf MZV$.