Multiple polylogarithms in weight 4
Abstract
We clarify the relationship between different multiple polylogarithms in weight~4 by writing suitable linear combinations of a given type of iterated integral In1,...,nd(z1,...,zd), in depth d>1 and weight Σi ni=4 in terms of the classical tetralogarithm Li4. In the process, we prove a statement conjectured by Goncharov which can be rephrased as writing the sum of iterated integrals I3,1(V(x,y),z), where V(x,y) denotes a formal version of the five term relation for the dilogarithm, in terms of Li4-terms (we need 122 such).
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