Prove $\{(u^3,u^2v,uv^2,v^3)\mid u,v\in\mathbb C\}$ is an algebraic set.

Question

I have to prove that $V=\{(u^3,u^2v,uv^2,v^3)\mid u,v\in\mathbb C\}$ is an algebraic set. How can I find an ideal $I\leq \mathbb C[x,y,z,t]$ s.t. $$V=Z(I)=\{(x,y,z,t)\in \mathbb C^4\mid \forall f\in I, f(x,y,z,t)=0\}\ \ ?$$

I really have no idea.

Answer 1

The idea is to find polynomial relations satisfied by $x = u^3, y= u^2v, z = uv^2, t = v^3$. For instance, since $$ xt = u^3 v^3 = u^2 v \cdot u v^2 = yz $$ then one such relation is $xt - yz = 0$. There should be two more, both degree $2$ like this one: see if you can find them.

The other two quadrics are $x z - y^2$ and $yt - z^2$.

If you want a general systematic way of finding implicit equations for a parametrization, you can use Gröbner bases. This is discussed in $\S3.3$: Implicitization of Cox, Little, and O'Shea's Ideals, Varieties, and Algorithms. See this post for more details.