Solutions of equations involving the modular $j$ function

TL;DR

This paper investigates solutions to systems of equations involving the modular j function, establishing conditions for solutions and linking the modular Schanuel conjecture to the existence of generic solutions, with some unconditional results.

Contribution

It provides new insights into solutions of modular j function equations, connecting them to the modular Schanuel conjecture and proving unconditional results for specific polynomial cases.

Findings

Certain systems of j equations have solutions under general conditions.

The modular Schanuel conjecture implies the existence of generic solutions.

Unconditional solutions are proven for polynomial equations with algebraic coefficients.

Abstract

Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions, and then we look at certain situations in which the modular Schanuel conjecture implies that these systems have generic solutions. An unconditional result in this direction is proven for certain polynomial equations on $j$ with algebraic coefficients.