2020/12/10 by Daniil Rudenko, Rudenko, Daniil · 1 citation
Mathematics · Chemistry · #Advanced Combinatorial Mathematics #Analytic and geometric function theory #Molecular spectroscopy and chirality
paper · pdf · doi:10.48550/arxiv.2012.05599
We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that correspond to decompositions of a polygon into quadrangles. Our second result is a formula for volume of hyperbolic orthoschemes, generalizing the formula of Lobachevsky in dimension 3 to an arbitrary dimension. We show a surprising relation between two results, which comes from the fact that hyperbolic orthoschemes are parametrized by configurations of points on ℙ1. In particular, we derive both formulas from their common generalization.