Show that a complex polynomial $P$ in $\mathbb C^n$ is identically zero if $\int_{T^n} |P|\, d\sigma = 0$

Question

Chapter 8.

1. (b) If $P$ is a polynomial in $\mathbb{C}^n$ and if $$ \int_{T^n} \vert P\vert \,d\sigma_n = 0 $$ then $P$ is identically $0$. (Hint: Compute $\int_{T^n} \vert P\vert^2 \,d\sigma_n$).

Here, $T^n$ is the torus that consists of all points $w = (e^{i\theta_1}, \ldots, e^{i\theta_n})$, and $\sigma_n$ is a Haar measure on it.

I think since $\vert P\vert$ is continuous on $T^n$ from $\int_{T^n} \vert P\vert \,d\sigma_n = 0 $, $\vert P\vert = 0$ on $T^n$, and accordingly, $\int_{T^n} \vert P\vert^2 \,d\sigma_n = 0$. However, how can I proceed along? Any help or hint would be appreciated!

Answer 1

I just give you the idea.

Once you know $\int |P^2|=0$, write $|P|^2$ as $P . \bar{P}$ (the complex conjugation) and express $P= \sum a_{i_1,\cdots,i_n} X_1^{i_1} \cdots X^{i_n}$. Developing and expressing integration on $T^n$ will give you a sum of terms $\int_{[0,2 \pi]^n} a_{i_1,\cdots,i_n} \bar{a}_{j_1,\cdots,j_n} e^{i(i_1-j_1) t_1+\cdots + i(i_n-j_n)t_n} dt_1 \cdots dt_n$. You can easily calculate these terms and you should be able to conclude.