I want to compute the Zariski closure of $Z = \lbrace (t, t^2, t^3) \mid t \in \mathbb{C} \rbrace \subseteq \mathbb{A}^3(\mathbb{C})$. I know that for this, it suffices to compute $V(\operatorname{Ann}(Z))$, where $\operatorname{Ann}(Z)$ is the annihilator of $Z$ and $V$ is the vanishing locus.
I think $\operatorname{Ann}(Z)$ is $(x^2-y,x^3-z)$. In any case, $(x^2-y,x^3-z) \subseteq \operatorname{Ann}(Z)$. The reverse inclusion is what is giving me trouble: My reasoning would be that if $f \in \operatorname{Ann}(Z)$, then $f(a, b, c) = 0$ if $b = a^2$ and $c=a^3$, so $x^2-y$ and $x^3-z$ divide $f$. But I'm not sure if this is sufficient.
Intuitively, $Z$ should already be closed, and if I computed the annihilator correctly, this should indeed be the case. I'm just unsure of whether I took the right steps, as I am not particularly comfortable with polynomials in multiple variables.