# If P is a polynomial in CC^n and if int_(T_n) |P|d sigma_n=0, then P is identically zero.

If P is a polynomial in ${\mathbb{C}}^{n}$ and if
${\int }_{{T}^{n}}|P|d{\sigma }_{n}=0,$
then P is identically zero.
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Emilia Boyle
If you restrict P as a function ${\mathbb{T}}^{n}$, then the Fourier series for P is simply the polynomial itself (up to constant maybe). Therefore, by Plancherel's theorem, we have
${\int }_{{\mathbb{T}}^{n}}|P|=0⟹{\int }_{{\mathbb{T}}^{n}}|P{|}^{2}=\sum _{j}|{a}_{j}{|}^{2}=0$
again with possibly a constant $C>0$ in front, and the ${a}_{j}$'s are the Fourier coefficients of P, which are the coefficients of the polynomial! Therefore ${a}_{j}=0$ for all j, hence $P\equiv 0$