Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint.

It is not difficult to see that $$\int_{1<|z|<2}\prod_{\{i,j\}\in E}\left(\frac{z_i}{z_j}+\frac{z_j}{z_i}-2\right)dz\neq0$$ where $|z|=\sqrt{|z_1|^2+\cdots+|z_n|^2}$ and $dz$ denotes the Lebesgue measure on $\mathbb{C}^n$.

My question is whether it is possible that $$\int_{1<|z|<2}\prod_{\{i,j\}\in E}\left(\frac{z_i}{z_j}+\frac{z_j}{z_i}-2\right)\prod_{\{i,j\}\in F}|z_i-z_j|^2dz=0.$$ I would be very grateful for thoughts on how to tackle this problem. Any idea is welcome!


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