Suppose that $I$ is a homogeneous ideal in $k[Z_0, Z_1, . . . , Z_n]$ such that the radical ideal $√I = (Z_0, . . . , Z_n)$.
Show that:
(a) There is an integer $N$ so that for any $d ≥ N$ and any homogeneous polynomial $f$
of degree $d$, then $f ∈ I$.
(b) Show conversely that if $I$ is a homogeneous ideal such that $V (I) = ∅∈ \mathbb{P}^
n$ then there
exists an $N$ with the property from (a).
Since $√I = (Z_0, . . . , Z_n)$ by definition of radical ideal we can say that for each $j = 0, . . . , n$ there is a positive integer $n_j$ so that $Z_j^{n_j}∈ I$
Is $N=max({n_j})$? Because any polynomial of degree $≥ N$ would be included in the ideal.
The variety in $I$, i.e. $V(I)$ is empty in $\mathbb{P}^n$ because it correponds to the point $(0,0,...,0)$ which is doesn't belong to $\mathbb{P}^n$. But how do we get (a) from that?