On Ecalle's and Brown's polar solutions to the double shuffle equations modulo products

TL;DR

This paper compares two explicit solutions to the double shuffle equations modulo products introduced by Ecalle and Brown, revealing their agreement up to depth four and differences at depth five with an explicit solution involving an exotic pole structure.

Contribution

The paper unifies Ecalle's and Brown's solutions within a common algebraic framework and explicitly characterizes their divergence at depth five.

Findings

Solutions agree up to depth four

Differences at depth five involve an exotic pole structure

Explicit solution to linearized equations identified

Abstract

Two explicit sets of solutions to the double shuffle equations modulo products were introduced by Ecalle and Brown respectively. We place the two solutions into the same algebraic framework and compare them. We find that they agree up to and including depth four but differ in depth five by an explicit solution to the linearized double shuffle equations with an exotic pole structure.