A review of Dan's reduction method for multiple polylogarithms

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 $\delta$.