Submission date: 15 December 2023

Abstract:

We show that for any polynomial F(X,Y_0,Y_1,Y_2) \in ℂ[X, Y_0, Y_1, Y_2], the equation F(z,j(z),j'(z),j''(z))=0 has a Zariski dense set of solutions in the hypersurface F(X,Y_0,Y_1,Y_2)=0, unless F is in ℂ[X] or it is divisible by Y_0, Y_0-1728, or Y_1.
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.

Mathematics Subject Classification: 11F03, 11F23, 11U09

Keywords and phrases:

Full text arXiv 2312.09974: pdf, ps.