How can I find a finite generating set for the ideal $I \subset \mathbb C[x,y]$ generated by the set $\{y -x,y - x^2, y-x^3,\dots\}$?

Question

How can I find a finite generating set for the ideal $I \subset \mathbb C[x,y]$ generated by the set $\{y -x,y - x^2, y-x^3,\dots\}$

I know $\mathbb C[x,y]$ is noetherian by the Hilbert Basis Theorem, and so $I$ is finitely generated, but how do I go about finding a finite generating set?

Any hints?

Answer 1

HINT:

The first two already generate the whole ideal. To see that look at the differences.