Notes on Ecalle's and Brown's solutions to the double shuffle reltions modulo products
Hidekazu Furusho, Minoru Hirose, Nao Komiyama
TL;DR
This work bridges Brown's and Ecalle's formalisms for the double shuffle relations modulo products by reinterpreting Brown's polar solutions $\psi_{2n+1}$ and $\psi_{-1}$ within Ecalle's mould theory, and by casting Brown's polynomial solution $\sigma_{2n+1}^{c}$ in the same mould-theoretic language as the Ecalle construct $\mathsf{luma}_{2n+1}$. It proves that $\psi_{2n+1}^{\sharp}=\mathsf{sang}(\mathsf{sa}_{2n+1})$ and characterizes $\psi_{-1}^{\#}$, thereby relating Brown's polar data to mould-theoretic operations. In the polynomial sector, the paper shows that the first depth-1/2 terms align ($\xi_{2n+1}=\mathsf{slang}_{1}(\mathsf{sa}_{2n+1})$ modulo higher depth) and identifies a depth-3 discrepancy between $\sigma_{2n+1}^{c}$ and $\mathsf{luma}_{2n+1}$, explicitly expressed via Bernoulli numbers and a polynomial $D_{a,b}$. These results provide a concrete dictionary between Brown's and Ecalle's constructions, clarifying how the two frameworks encode the same underlying double shuffle structure and highlighting where their polynomial data diverges at depth 3. The findings advance the program of understanding motivic and mould-theoretic aspects of MZV relations and offer practical tools for translating results across the two viewpoints.
Abstract
We investigate relationships between polar/polynomial solutions to the double shuffle relations modulo products, which were independently introduced by Brown and Ecalle.