Notes on Ecalle's and Brown's solutions to the double shuffle relations modulo products

TL;DR

This paper explores the connections between two types of solutions to the double shuffle relations modulo products, as introduced by Brown and Ecalle, aiming to deepen understanding of their mathematical structure.

Contribution

It provides a comparative analysis of Brown's and Ecalle's solutions, clarifying their relationships and implications for the double shuffle relations.

Findings

Identifies key similarities and differences between the solutions

Establishes new links between the two solution frameworks

Enhances understanding of the algebraic structure of the relations

Abstract

We investigate relationships between polar/polynomial solutions to the double shuffle relations modulo products, which were independently introduced by Brown and Ecalle.