I want to understand how I can find the intersection multiplicity $I_p$ at a point $p$ for two curves $f,g$.
I have the example where $$ f(x,y) = y^2-x^3, \,\,\,\, g(x,y)=y^2-x^2(x+1) $$ Then I am looking for the intersection multiplicity of the ideal $(f,g)$ at $p=(0,0)$.
My first attempt is to notice that $g(x,y)=y^2-x^3-x$ and the ideal $(f,g)$ can be written as $(f,g-f)=(y^2-x^3,x^2)$. But now I am stuck since I cannot find a way to possibly simplify this ideal as to conclude about the intersection form. Only step I can go further is to note that
$$ I_{(0,0)}(y^2-x^3,x^2) = 2I_{(0,0)}(y^2-x^3,x) $$
What would the next step be in order to determine $I_p$? In a previous example I was able to reduce the original ideal to an ideal involving only degree 1 curves concluding easily about what the multiplicity is. Now?