Polylogarithmic functions with prescribed branching locus and linear relations between them

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.