Arborified multiple zeta values

TL;DR

This paper explores a novel algebraic framework called arborification that connects multiple zeta values with decorated rooted forests, providing new insights into their algebraic structure and relations.

Contribution

It introduces a formalism linking multiple zeta values to decorated rooted forests via Hopf algebra morphisms, unifying sum and integral perspectives.

Findings

Establishes a surjective Hopf algebra morphism from forests to shuffles

Provides a unified algebraic description for multiple zeta values

Connects arborification with both sum and integral representations

Abstract

We describe some particular finite sums of multiple zeta values which arise from J. Ecalle's "arborification", a process which can be described as a surjective Hopf algebra morphism from the Hopf algebra of decorated rooted forests onto a Hopf algebra of shuffles or quasi-shuffles. This formalism holds for both the iterated sum picture and the iterated integral picture. It involves a decoration of the forests by the positive integers in the first case, by only two colours in the second case.