  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.
