Equations involving the Modular j-function and its derivatives

We show that, for any polynomial F(X, Y₀, Y₁, Y₂) ∈ ℂ[X, Y₀, Y₁, Y₂], the equation F(z, j(z), j′(z), j′′(z)) = 0 has a Zariski dense set of solutions in the hypersurface F(X, Y₀, Y₁, Y₂) = 0, unless F is in ℂ[X] or it is divisible by Y₀, Y₀ − 1728, or Y₁. Our methods establish criteria for finding solutions to more general equations involving periodic functions. Furthermore, they produce a qualitative description of the distribution of these solutions.