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# Calculating the local index of intersection of two algebraic curves.

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?

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# Calculating local index of intersection of two algebraic curves.

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 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 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 $\mathbb C^2$ 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?