Equations involving the modular j-function and its derivatives

2023/12/15 by Vahagn Aslanyan, Aslanyan, Vahagn, Sebastian Eterović +3 · 1 citation
Mathematics · #11F03 #11F23 #11U09 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2312.09974

openalex publication_date 2023/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

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.

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