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Iterated Integrals and Multiple Polylogarithm at Algebraic Arguments

Kam Cheong Au

TL;DR

This work develops a unified framework linking cyclotomic MZVs to generalized iterated integrals over arbitrary finite sets on the Riemann sphere, enabling CMZVs to be realized as $\textsf{MZV}^{S}_{n}$ for suitable $S$ and revealing robust transformation properties under rational maps. The core result equates CMZV$^N_n$ with $\textsf{MZV}^{S}_{n}$ for $S=\{0,\infty,1,\mu,\dots,\mu^{N-1}\}$, providing a powerful tool to generate CMZVs, prove non-standard relations via $S$-unit equations, and compile a datamine together with a computational package. The authors establish a regularized, path-aware framework leveraging tangential base points, enabling explicit computations, descent conjectures, and applications to Apéry-type series, while also exploring polylogarithm identities and special values through Coxeter ladders and icosahedral symmetry. Collectively, these contributions yield new CMZV relations, scalable computational resources, and a versatile perspective on the arithmetic of polylogarithms with algebraic arguments. The approach broadens access to CMZV identities and offers practical mechanisms for verifying and discovering relationships among polylogarithm values in number theory and related fields.

Abstract

By introducing a generalized notion of multiple zeta values associated with an arbitrary finite subset $S\subset \mathbb{P}^1(\mathbb{C})$ and studying their transformation properties under rational functions, we show that multiple polylogarithms evaluated at roots of unity (cyclotomic multiple zeta values, CMZVs) can be equivalently expressed in terms of iterated integrals involving certain non-roots of unity. We apply this theory to elucidate previously unknown $\mathbb{Q}$-linear relations among CMZVs: they come from nontrivial solutions of certain $S$-unit equations in the function field of $\mathbb{P}^1(\mathbb{C})$, thereby attaining the motivic dimension for low level and weight. We introduce a datamine of CMZVs that appears to be the first rigorous compilation of this kind in the literature. In addition, we formulate several nontrivial Galois descent conjectures for multiple polylogarithms and present applications to certain Apéry-type infinite series.

Iterated Integrals and Multiple Polylogarithm at Algebraic Arguments

TL;DR

This work develops a unified framework linking cyclotomic MZVs to generalized iterated integrals over arbitrary finite sets on the Riemann sphere, enabling CMZVs to be realized as for suitable and revealing robust transformation properties under rational maps. The core result equates CMZV with for , providing a powerful tool to generate CMZVs, prove non-standard relations via -unit equations, and compile a datamine together with a computational package. The authors establish a regularized, path-aware framework leveraging tangential base points, enabling explicit computations, descent conjectures, and applications to Apéry-type series, while also exploring polylogarithm identities and special values through Coxeter ladders and icosahedral symmetry. Collectively, these contributions yield new CMZV relations, scalable computational resources, and a versatile perspective on the arithmetic of polylogarithms with algebraic arguments. The approach broadens access to CMZV identities and offers practical mechanisms for verifying and discovering relationships among polylogarithm values in number theory and related fields.

Abstract

By introducing a generalized notion of multiple zeta values associated with an arbitrary finite subset and studying their transformation properties under rational functions, we show that multiple polylogarithms evaluated at roots of unity (cyclotomic multiple zeta values, CMZVs) can be equivalently expressed in terms of iterated integrals involving certain non-roots of unity. We apply this theory to elucidate previously unknown -linear relations among CMZVs: they come from nontrivial solutions of certain -unit equations in the function field of , thereby attaining the motivic dimension for low level and weight. We introduce a datamine of CMZVs that appears to be the first rigorous compilation of this kind in the literature. In addition, we formulate several nontrivial Galois descent conjectures for multiple polylogarithms and present applications to certain Apéry-type infinite series.
Paper Structure (25 sections, 33 theorems, 204 equations, 6 figures, 1 table)

This paper contains 25 sections, 33 theorems, 204 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology, main result and some consequences
  3. Non-standard relations
  4. Datamine of CMZVs
  5. Relations between polylogarithm and Galois descent
  6. Apéry-like series
  7. Preliminaries
  8. Shuffle algebra
  9. Iterated integral
  10. Tangential base point
  11. Iterated integral over general base
  12. Definition and main properties
  13. Consequences of the Main Theorem \ref{['CMZV_main_theorem']}
  14. Proof of Main Theorem \ref{['CMZV_main_theorem']}
  15. Explicit computations
  16. ...and 10 more sections

Key Result

Corollary 1.1

Let $S = \{0,\infty,1,e^{2\pi i/N},\cdots,e^{2\pi i (N-1)/N}\}$, $R(x)$ be a rational function such that $R^{-1}(R(S)) = S$ and $\{0,1,\infty\} \subset R(S)$, then the iterated integral provided that $c_1\neq 1, c_n\neq 0, c_i\in R(S).$

Figures (6)

  • Figure 1: Integration along loop enclosing $a_2$ and based at $a_1+\varepsilon$.
  • Figure 2: The path $\iota(\varepsilon)$. Here the paths $g_i\gamma_i$ are actually the paths $g_i\gamma_i$ restricted to interval $[\varepsilon,1-\varepsilon]$
  • Figure 3: Deforming the integration path $R^{-1}[0,1]$ to two parts: $g[0,1]^{-1}$ and $[0,1]$.
  • Figure 4: The paths $R_1\circ [0,1]$ and $T_1\circ [0,1]$.
  • Figure 5: The paths $R_1\circ [0,1]$ and $T_1\circ [0,1]$.
  • ...and 1 more figures

Theorems & Definitions (99)

  • Corollary 1.1
  • Theorem 1.2: Main theorem
  • Theorem 1.3
  • Definition 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 89 more