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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 $r$?

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    I think the first method to compute intersection multiplicities for all plane curves over the complex numbers was that of resultants. You take the $X$ resultant of 2 polynomials in $X$ and $Y$ which is a polynomial in $Y$, and I think the degree of this polynomial is your (total) intersection multiplicity. This is the appropriate (keeping multiplicities) projection of your intersection of the 2 varieties to 1 dimension, and the number of solutions (with multiplicities) in 1 dimension is easy to compute. http://en.wikipedia.org/wiki/Resultant . Wiki is not extremely helpful here.2012-05-22

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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 $2$.

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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).