Let $F_1,F_2$ be two polynomials of two variables $(x,y)$ (say complex variables). Suppose that $F_1$ and $F_2$ have no common factors and $F_1(P)=F_2(P)=0$.

What is in practice the quickest way to calculate the index of intersection of the curves $F_1=0$ and $F_2=0$ at $P$? Or, say, what methods one uses to calculate the index?

I know two definitions of the index but they don't look so handy if $C_1$ and $C_2$ are "complicated" at $P$:

1) Take generic small numbers $c_1,c_2\in \mathbb C$ and count the number of intersections of curves $F_1=c_1$, $F_2=c_2$ close to the point $P$.

2) Calculate the dimension of the vector space $O_P/((F_1,F_2)\cdot O_P)$ where $O_P$ denotes the local ring of $\mathbb C^2$ at $P$.

I guess, there should be one more way to make the calculation, by resolving singularities of curves $C_1$ and $C_2$ at $P$...

But how to do this calculation in an effective way?


The index of intersection satisfies certain properties which are easier to apply than the two definitions you give. For instance, if the tangent cones at P (initial forms, if $P=(0,0)$) of $F_1$, $F_2$ have no common factors, the index of intersection is just the product of multiplicities at $P$. And the index of intersection of $F_1=0$, $F_2=0$ coincides with the index of intersection of $F_1=0$ and $F_2+G\cdot F_1=0$ for every $G$. You can find a list of the relevant properties, with examples on how to apply them, in the relevant section of Fulton's book on Algebraic Curves.

There are two other ways to compute this number. One is, as you say, resolve the singularities of the union $F_1\cdot F_2=0$; then the intersection index is the sum of the products of the multiplicities of (the strict transforms of) $F_i=0$ at all blown up points (this is due to Max Noether). The other is to parameterize all branches of one of the curves $F_1=0$ and substitute the parameterizations in the other equation $F_2=0$. If the parameterizations are minimal, the intersection index is the sum of the resulting orders. Both these ways are explained in Casas-Alvero's book on Singularities of Plane Curves.

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