Exercise from P.159 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide.

Given affine curves $x = y^2$ and $y = −3$, find the points of intersection of the corresponding projective curves.

After homogenization, we obtain:

  • $zx = y^2$
  • $y = -3z$.

We split this into two cases:

  • $z = 0$. Then $y$ must be $0$, but $x$ does not have to be. Therefore, $[1, 0, 0]$ is a solution. Indeed, for any $t \in \mathbb{R}$, $(t, 0, 0)$ satisfies the system of equations.
  • $z \ne 0$. Without loss of generality, set $z = 1$. Then we obtain $[9, -3, 1]$. Indeed for any $t \in \mathbb{R}$, $(9t, -3t, t)$ is a solution.

Therefore, the corresponding projective curves intersect at $[1, 0, 0]$ and $[9, -3, 1]$.