Equations involving the modular $j$-function and its derivatives

We show that for any polynomial $F(X,Y_0,Y_1,Y_2)\in \mathbb{C}[X,Y_0,Y_1,Y_2]$, the equation $F(z,j(z),j^{\mathrm{\prime }}(z),j^{\mathrm{\prime }\mathrm{\prime }}(z))=0$ has a Zariski dense set of solutions in the hypersurface $F(X,Y_0,Y_1,Y_2)=0$, unless $F$ is in $\mathbb{C}[X]$ 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.