Polylogarithmic functions with prescribed branching locus and linear relations between them

TL;DR

This paper develops an algorithm to identify all classical polylogarithmic functions with a specific branching locus defined by polynomial solutions, enabling the discovery of new identities among polylogarithms.

Contribution

It introduces a novel algorithm for constructing all possible arguments of polylogarithms with prescribed branching loci, supported by computational implementation.

Findings

Algorithm successfully constructs complete sets of polylogarithm arguments.

Provides new examples of polylogarithmic identities.

Includes a Mathematica implementation for practical use.

Abstract

We consider the problem of finding the set of classical polylogarithmic functions $\text{Li}_n$ with branching locus determined by the solution of $p_1\cdot p_2\cdot \ldots \cdot p_n=0$, where $p_1,\ldots, p_n$ are irreducible polynomials of several variables. We present an algorithm of constructing a complete set of possible arguments of $\text{Li}_n$ functions. The corresponding Mathematica code is included as ancillary file. Using this algorithm and the symbol map, we provide some examples of polylogarithmic identities.