I want to compute the intersection multiplicity of $YZ=X^2$ and $YZ=(X+Z)^2$ at $P=(0:1:0)$ In an affine nbd of $P$, let $(X:Y:Z)=(x:1:z)$ $I_P=\dim \mathcal{O}_{\mathbb{A}_k^2,(0,0)}/(x^2-z,x^2+2xz+z^2-z)$
In my previous question, I learned that if $(0,0)$ is the only zero, I can change the local ring to $k[x,z]$. But the problem occurs since there are intersection points not only $(0,0)$, but also $(-2,4)$. I saw the Fulton's book(section 2.9 prop 6), which says that $k[x,z]/I \simeq \mathcal{O}_{(0,0)}/I\mathcal{O}_{(0,0)} \times \mathcal{O}_{(-2,4)}/I\mathcal{O}_{(-2,4)}$ So now how can I compute the intersection multiplicity?