Proposition

Show that $\phi_1(x, y, z) = (2x^2 + y^2, z^2 - y^3 + 3xz)$ and $\phi_2(x, y, z) = (2y + xz, 3y^2)$ represent the same polynomial mapping from the twisted cubic in $\mathbb{R}^3$ to $\mathbb{R}^2$.

Solution

$\{ (a, a^2, a^3) \mid a \in \mathbb{R} \} \subset \mathbb{R}^3$ is the twisted cubic. $\phi_1(a, a^2, a^3) = (2a^2 + a^4, 2a^4) = \phi_2(a, a^2, a^3)$, so they represent the same polynomial mapping.