Counting multiplicities and Bezout's theorem What is the method to count multiplicities of intersection? for example suppose we have the projective line $x=0$ in $\mathbb{P}^{2}$ and the curve $V(z^{2}y^{2}-x^{4}) \subseteq \mathbb{P}^{2}$.
Clearly they intersection consists of two points $p=[0:1:0]$ and $r=[0:0:1]$. So for example Bezout's theorem says that the sum of intersection multiplicities (at $p$ or $r$) is equal to $4$. Is there a way to know exactly what is the multiplicity of $p$ and $r$? 
 A: Sure. Theorem 3 of Section 3.3 of Fulton's Algebraic Curves gives an algorithm for computing the intersection number at a point P.
For the projective case you need to dehomogenize with respect to the "proper" line to reduce it to the affine case.
There are some examples here (Q5-3).
A: For the intersection of two plane curves (for simplicity, assumed to be defined over $\mathbb{C}$) at a point $P$, the multiplicity is computed as follows.  Let $f$ be a local equation for one curve and $g$ a local equation for the other curve near the point $P$.  Then multiplicity of $P$ in the intersection of the two curves is the dimension of the $\mathbb{C}$-vector space $\mathcal{O}_P / (f,g)$.  Here $\mathcal{O}_P$ is the local ring of the plane at $P$ and $(f,g)$ is the ideal generated by $f$ and $g$.
In your example, consider the point $p = [0:1:0]$.  We can work locally in the polynomial ring $\mathbb{C}[x,z]$, where $x = X/Y$ and $z = Z/Y$.  (This is not quite the local ring $\mathcal{O}_P$ referred to above, but we can use it because it is the ring of functions in a neighborhood of $p$ that does not contain $r$.)  The local equations for your curves are $f = x$ and $g = z^2 - x^4$.  Thus, we must compute the dimension of $\mathbb{C}[x,z] / (x, z^2 - x^4)$, which has basis $\{1, z\}$.  Thus the multiplicity of $p$ is $2$.
