On The Global Renormalization and Regularization of Several Complex Variable Zeta Functions by Computer

TL;DR

This paper reviews the use of computer-aided methods to analyze and solve complex variable zeta functions related to the Knizhnik-Zamolodchikov equations, focusing on the Drinfel'd associator and related series.

Contribution

It introduces a computational approach to resolve specific KZ equations, providing explicit solutions and algebraic structures for complex zeta functions.

Findings

Explicit solution for the KZ_3 equation leading to the Drinfel'd associator

Derivation of non-trivial series with rational coefficients

Analysis of algebraic structures and singularities of polylogarithms

Abstract

This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations ($KZ_3$) using our recent results on combinatorial aspects of zeta functions on several variables and software on noncommutative symbolic computations. In particular, we describe the actual solution of $(KZ_3)$ leading to the unique noncommutative series, $\Phi_{KZ}$, so-called Drinfel'd associator (or Drinfel'd series). Non-trivial expressions for series with rational coefficients, satisfying the same properties with $\Phi_{KZ}$, are also explicitly provided due to the algebraic structure and the singularity analysis of the polylogarithms and harmonic sums.