how do we decompose the following algebraic sets :
- $V(x^3-2x^2y+2xy^2-y^3, x^4-x^3y-xy+y^2) \subseteq \mathbb{C}^2$ and
- $V(2x^3z+3x^2y^2+6y^3+4xyz+z^2) \subseteq F^2$ where $F$ is algebraically closed.
into irreducible components?
For ideals of the form $V(f)$ where $f$ is some simple polynomials, say, $f$ is in $F[x]$ or $f$ does not involve too many terms of high powers, I know how to decompose that kind of algebraic sets.
But if $f$ has the above form, how do I decompose $V(f)$? Is there a general method for this type of problem?