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On the decomposition of motivic multiple zeta values

Francis Brown

TL;DR

The article develops a motivic framework for multiple zeta values (MZVs), treating them as elements of a coalgebra that admits a coaction from a motivic Galois group. It introduces a concrete, exact, and numerically implementable decomposition algorithm that expresses any motivic MZV in terms of a chosen basis by using derivations ∂_{2n+1} and a normalization φ that maps motivic zetas to noncommutative generators, reducing the problem to one-dimensional lattice reductions. A central tool is the coaction structure and the associated derivations that allow one to extract basis coordinates in a weight-by-weight induction, with explicit worked examples up to weight 10. The approach not only provides a practical method for decompositions but also offers a mechanism to lift relations between real MZVs to motivic level and to study the algebraic structure of MZVs through the motivic lens, including the role of zeta cogenerators and basis choices such as Hoffman-type bases. The work thereby bridges iterated-integral representations, motivic formalism, and computational algorithms, with potential applications in mathematical physics and number theory.

Abstract

We review motivic aspects of multiple zeta values, and as an application, we give an exact-numerical algorithm to decompose any (motivic) multiple zeta value of given weight into a chosen basis up to that weight.

On the decomposition of motivic multiple zeta values

TL;DR

The article develops a motivic framework for multiple zeta values (MZVs), treating them as elements of a coalgebra that admits a coaction from a motivic Galois group. It introduces a concrete, exact, and numerically implementable decomposition algorithm that expresses any motivic MZV in terms of a chosen basis by using derivations ∂_{2n+1} and a normalization φ that maps motivic zetas to noncommutative generators, reducing the problem to one-dimensional lattice reductions. A central tool is the coaction structure and the associated derivations that allow one to extract basis coordinates in a weight-by-weight induction, with explicit worked examples up to weight 10. The approach not only provides a practical method for decompositions but also offers a mechanism to lift relations between real MZVs to motivic level and to study the algebraic structure of MZVs through the motivic lens, including the role of zeta cogenerators and basis choices such as Hoffman-type bases. The work thereby bridges iterated-integral representations, motivic formalism, and computational algorithms, with potential applications in mathematical physics and number theory.

Abstract

We review motivic aspects of multiple zeta values, and as an application, we give an exact-numerical algorithm to decompose any (motivic) multiple zeta value of given weight into a chosen basis up to that weight.

Paper Structure

This paper contains 26 sections, 6 theorems, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Iterated Integrals
  3. General iterated integrals.
  4. The punctured projective line.
  5. Multiple zeta values.
  6. Regularization of MZVs
  7. Structure of MZV's in low weights
  8. Motivic formalism
  9. The category of mixed Tate motives over $\mathbb Z$
  10. Motivic multiple zeta values.
  11. Notations
  12. Formula for the coaction
  13. Zeta cogenerators.
  14. The derivations $\partial_{2n+1}$
  15. Truncation operators on $\mathcal{U}$
  16. ...and 11 more sections

Key Result

Lemma 2.2

There is a unique way to define a set of real numbers $I(a_0;a_1,\ldots,a_n;a_{n+1})$ for any $a_i\in \{0,1\}$, such that

Figures (1)

  • Figure :

Theorems & Definitions (20)

  • Definition 2.1
  • Lemma 2.2
  • Remark 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Example 3.4
  • Conjecture 1
  • Theorem 3.5
  • Definition 3.6
  • ...and 10 more