Determine the cardinality of the set $ V (x^3-yz, y^2-xz, z^2-x^2y ) $ and compare this to the dimension of $ \dfrac{ \mathbb C [x,y,z]}{\langle x^3-yz, y^2-xz, z^2-x^2y \rangle} $ viewed as a complex vector space.
For the first part, I obtained a parametric family of solutions $ (x,y,z) = (t^3, t^4, t^5), \; t \in \mathbb C $ so the vanishing set is infinite. I believe the answer to the second part is infinite but I'm not sure how to verify this in the given case (so far, I've only encountered the quotient of a polynomial ring in a single variable)