This is Exercise 1 of Chapter 8 in Rudin's Functional Analysis. We are asked to prove the following:
If $P$ is a polynomial in $\mathbb{C}^n$ and if \begin{equation}\int_{T^n}|P|d\sigma_n=0,\end{equation} then $P$ is identically zero.
Here $T_n$ is the n-dimensional torus and $\sigma_n$ is the Haar measure on $T_n$.
It should not be difficult but I guess I am missing a trick in complex analysis.
Thanks!