Calculate the multiplicity at the origin of the intersection of the curves $X=V(xy^2+x^4)$ and $Y=V(x^2y-x^4)$.
This question was already posted here Calculate the multiplicity at the origin of the intersection of the curves $X=V(xy^2+x^4)$ and $Y=V(x^2y-x^4)$, but I would like to know how to solve this using that $I_P(X\cap Y)=\dim_k(\mathbb{O}_p(A^2)/(f,g))$? In this case what would $\mathbb{O}_p(A^2)$ and $(f,g)$ be? Thank you.
The notation $\Bbb{O}_p(A^2)$ is kind of funny - it's more usually rendered as $\mathcal{O}_{\Bbb A^2_k,p}$. This is the local ring of $\Bbb A^2_k$ at the point $p$ - in this case, it's $k[x,y]_{(x,y)}$. $(f,g)$ is the ideal of $\mathcal{O}_{\Bbb A^2_k,p}$ generated by $f$ and $g$: in this case, $f=xy^2+x^4$ and $g=x^2y-x^4$. From here you're just computing the dimension of the quotient as a $k$-algebra, which should be straightforwards.