ON ECALLE'S AND BROWN'S POLAR SOLUTIONS TO THE DOUBLE SHUFFLE EQUATIONS MODULO PRODUCTS

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.