Evaluating Multiple Polylogarithm Values at Sixth Roots of Unity up to Weight Six

TL;DR

The paper advances the computation of multiple polylogarithms evaluated at sixth roots of unity up to weight six by constructing explicit bases for the real and imaginary parts of $G(a_1,\,\ldots,\,a_w;1)$ and expressing all values as rational combinations of basis elements. It systematically applies shuffle, stuffle, regularization, and distribution relations and uses PSLQ to prune dependencies, delivering a comprehensive reduction table and basis definitions up to weight 6. The work confirms dimensional and structural conjectures from Deligne–Goncharov motivic theory and Broadhurst, and it provides practical, downloadable resources for high-precision multiloop Feynman integral calculations. These results enable efficient evaluation of MPL constants appearing in three-loop massive form factors and related problems in quantum field theory. The combination of recursive basis construction, extensive relation handling, and PSLQ-based pruning highlights both the mathematical structure and the computational approach needed at this level of weight and root-of-unity alphabet.

Abstract

We evaluate multiple polylogarithm values at sixth roots of unity up to weight six, i.e. of the form $G(a_1,\ldots,a_w;1)$ where the indices $a_i$ are equal to zero or a sixth root of unity, with $a_1\neq 1$. For $w\leq 6$, we present bases of the linear spaces generated by the real and imaginary parts of $G(a_1,\ldots,a_w;1)$ and present a table for expressing them as linear combinations of the elements of the bases.