On the functional equations for polylogarithms in one variable

Abstract: We develop a new approach to the study of the functional equations satisfied by classical polylogarithms, inspired by Goncharov's conjectures. We prove a sharpened version of Zagier's criterion for such an equation and explain, how our approach leads to a very simple description of the equations in one variable, satisfied by dilogarithm and trilogarithm. Our main result is the complete description of the functional equations for weight four polylogarithm in one variable.