# On generic double shuffle relations, localized multiple polylogarithms and algebraic functions

## TL;DR

This paper introduces algebraic functions on subvarieties of moduli spaces that solve the double shuffle equations characteristic of multiple polylogarithms, advancing understanding of their algebraic structure.

## Contribution

It defines new subvarieties of _{n} with algebraic functions satisfying the double shuffle relations of multiple polylogarithms, linking geometric and algebraic aspects.

## Key findings

- Identification of subvarieties with algebraic solutions to double shuffle equations
- Establishment of a geometric framework for multiple polylogarithms
- Connection between algebraic functions and moduli space structures

## Abstract

We define subvarieties of $\mathcal{M}_{0,n}$ equipped with algebraic functions that are solutions to the generic double shuffle equations satisfied by multiple polylogarithms on $\mathcal{M}_{0,n}$.