Solutions of equations involving the modular $j$ function

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.

Differential Existential Closedness for the $j$-function

“…Both of those are analogues of Zilber's conjecture mentioned above. Eterović and Herrero have recently made progress towards the complex EC for the j-function [EH20].…”