A review of Dan's reduction method for multiple polylogarithms

Abstract

In this paper we will give an account of Dan's reduction method for reducing the weight $ n $ multiple logarithm $ I_{1,1,\ldots,1}(x_1, x_2, \ldots, x_n) $ to an explicit sum of lower depth multiple polylogarithms in $ \leq n - 2 $ variables. We provide a detailed explanation of the method Dan outlines, and we fill in the missing proofs for Dan's claims. This establishes the validity of the method itself, and allows us to produce a corrected version of Dan's reduction of $ I_{1,1,1,1} $ to $ I_{3,1} $'s and $ I_4 $'s. We then use the symbol of multiple polylogarithms to answer Dan's question about how this reduction compares with his earlier reduction of $ I_{1,1,1,1} $, and his question about the nature of the resulting functional equation of $ I_{3,1} $. Finally, we apply the method to $ I_{1,1,1,1,1} $ at weight 5 to first produce a reduction to depth $ \leq 3 $ integrals. Using some functional equations from our thesis, we further reduce this to $ I_{3,1,1} $, $ I_{3,2} $ and $ I_5 $, modulo products. We also see how to reduce $ I_{3,1,1} $ to $ I_{3,2} $, modulo $ \delta $ (modulo products and depth 1 terms), and indicate how this allows us to reduce $ I_{1,1,1,1,1} $ to $ I_{3,2} $'s only, modulo $ \delta $.