Decomposition into irreducible algebraic sets

Question

I am facing following problem and would really appreciate anyone's help:

I am to find decomposition of $V(x^2 - y^4, x^3 - xy^2 + x^2y^2 -y^4)$ into irreducible algebraic sets in $A^2( \mathbb{C})$. I went for obvious parametrization, and from the first part got that $x=t^2$, $y=t$ or $y=-t$, second part than gives $t^6 = t^4$. Seems like the solutions are only points $(0,0), (1,1), (1,-1)$. So...is this the decomposition I am looking for just these three points, or am I completely wrong?

Thanks for any input, I am quite new to algebraic geometry and really confused.

Answer 1

First, you need to factor both polynomials.Then you pair all factors of one with all factors of the other to get a number of varieties. It remains then to show that these resulting varieties are irreducible, which can be done by showing the ideals are prime.You also want to check that this representations is irredundant, that no variety is contained in the union of the others.