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Multiple polylogarithms in weight 4
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We clarify the relationship between different multiple polylogarithms in weight~4 by writing suitable linear combinations of a given type of iterated integral I_{n_1,...,n_d}(z_1,...,z_d), in depth d>1 and weight \sum_i n_i=4 in terms of the classical tetralogarithm Li_4. In the process, we prove a statement conjectured by Goncharov which can be rephrased as writing the sum of iterated integrals I_{3,1}(V(x,y),z), where V(x,y) denotes a formal version of the five term relation for the dilogarithm, in terms of Li_4-terms (we need 122 such).
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On functional equations for Nielsen polylogarithms
The weight-5 Nielsen polylogarithm S3,2 satisfies the dilogarithm five-term relation modulo Li5 and products, with new explicit identities in weights up to 8.
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