 Graded weights separate analytic polynomial relations

Let $P_w(E_4,E_6)$ be weighted homogeneous of modular weight $w$, and suppose ${\sum }_w{P}_w(z)=0$ as a holomorphic function. Apply the modular group matrices $\left(\begin{array}{cc}1& 0\\ n& 1\end{array}\right)$. At a fixed point $z$ in the complex upper half-plane, this gives ${\sum }_w(nz+1{)}^w{P}_w(z)=0$ for every integer $n$. A polynomial with infinitely many roots vanishes identically, so every $P_w(z)=0$. Within one modular weight, the monomials reduce to powers of $E_4^3/E_6^2$, after taking out a common factor. This ratio is nonconstant, as its Fourier expansion begins $1+1728q+\cdots$, so its powers are linearly independent. This proves algebraic independence even when the generators are regarded as analytic functions rather than elements of a formal graded ring.