I was wondering if there are any results available on square integrability of quotients of polynomials $P(z_1,z_2,\ldots,z_n)$ and $Q(z_1,z_2,\ldots,z_n)$ like
$$\int_{\mathbb{T}^n} \left|\frac{P(z_1,z_2,\ldots,z_2)}{Q(z_1,\ldots,z_n)}\right|^2dz_1 \ldots dz_n,$$
where $\mathbb{T}$ is the region $|z_1|=1,\ldots,|z_n|=1$ and $Q$ has a zero on this region. I suppose if $P=1$, then the integral is infinity as in dimension one. I am not quite sure, how to prove it. I tried to use a Laurent expansion. My general problem is that there are examples like $P(z_1,z_2)=(1-z_1)(1-z_2)$ and $Q(z_1,z_2)=1-0.5z_1-0.5z_2)$ such that the integral exists, even though there is a singularity. Any suggestions?
Regards, Chesspert