= Graded weights separate analytic polynomial relations
Let $P_w(E_4,E_6)$ be weighted homogeneous of <modular weight> $w$, and suppose $\sum_wP_w(z)=0$ as a <holomorphic function>. Apply the <modular group> <matrices> $\begin{pmatrix}1&0\\n&1\end{pmatrix}$. At a fixed point $z$ in the <complex upper half-plane>, this gives $\sum_w(nz+1)^wP_w(z)=0$ 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>.
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