Let $\phi: \mathbb{R}^1 \longrightarrow \mathbb{R}^2$ be the map given by $t \mapsto (t^2,t^3)$. I'm trying to show that any polynomial $f \in \mathbb{R}[X,Y]$ vanishing on the image $C = \phi(\mathbb{R}^1)$ is divisible by $Y^2-X^3$. And what property of a field $k$ will ensure that the result holds for $\phi: k \longrightarrow k^2$ given by the same formula?
Also, I am trying to do it for $t \mapsto (t^2-1,t^3-t)$.
Thanks.