Finding cancelling polynomials of a set

Question

Let $V=\{(t,t^2,t^3),t \in \mathbb C\}$. Find $I(V)$. I found some polynomials that cancels in $V$. For example $X-Y^2$, $Y^2-Z^3$ or $X+Y^2-Z^3$ but I don't know how to find all polynomials, that is how to determine $I(V)$.

Thank you!

Answer 1

Consider the ideal $I=\langle x-t, y-t^2, z-t^3\rangle$ in ${\Bbb K}[x,y,z,t]$ with elimination order (lex) such that $t>x>y>z$. The first elimination ideal $I_1 = I\cap {\Bbb K}[x,y,z]$ is what you are looking for. You just need to construct the Groebner basis of $I$ and then take those generators that are polynomials in $x,y,z$. Done.

I'd consider the book of Cox et al., "Ideal, Varieties and Algorithms", Springer. Keyword: implicitization.