I'd like to prove the following:
If $\mathfrak{a} \subseteq k[x_0, \ldots, x_n]$ is a homogeneous ideal, and if $f \in k[x_0,\ldots,x_n]$ is a homogeneous polynomial with $\mathrm{deg} \ f > 0$, such that $f(P) = 0 $ for all $P \in Z(\mathfrak{a})$ in $\mathbb P^n$, then $f^q \in \mathfrak{a}$ for some $ q > 0$.
I've been given the hint: interpret the problem in terms of the affine ($n+1$)-space whose affine coordinate ring is $k[x_0,\ldots,x_n]$ and use the usual Nullstellensatz.
I'm not really sure what the hint means. We have the isomorphism $k[x_0,...,x_n] \cong k[x_0,...,x_n] / I(\mathbb A_k^{n+1})$ (since $I(\mathbb A^{n+1}) = I(Z(0)) = 0$). But I don't see how this is helpful at all, nor am I sure this is what the hint means.
Any help would be greatly appreciated. Thanks