Algebraic curves If I have two complex curves  $f = Y^2 + X^2Y - X^3$ and $g = XY^2 + 4Y^2 + X^3$ calculating their tangent in $ P = (0,0)$ this are $Y^2$ not? Well, so how I can calculate the number of intersection in $P$?
I don't know calculate it because the tangent of f and g are equal. 
 A: One way of doing this uses the fact that for every polynomial $h \in \Bbb{C}[X,Y]$ we have
$$
I(P,f \cap g)=I(P,(f+gh) \cap g)
$$
This allows you to simplify the polynomials you intersect, without changing the intersection number. For example, if $P=(0,0)$ you have
$$
I(P,(Y^2-X^3) \cap X)=I(P,Y^2 \cap X)=2
$$
where I set $h=X^2$.
So in your example above, a starting point could be
$$
\begin{align}
I(P,(Y^2+X^2Y-X^3) \cap (XY^2+4Y^2+X^3))& =I(P,(5Y^2+X^2Y+XY^2) \cap (XY^2+4Y^2+X^3)) \\
& =I(P,Y \cap (XY^2+4Y^2+X^3))\\
&+I(P,(5Y+X^2+XY) \cap (XY^2+4Y^2+X^3)) \\
& =...
\end{align}
$$
where in the last equality I have used multiplicativity of the intersection number.
I leave the rest of the computation to you, because if you don't do it yourself, you probably won't learn it. But feel free to ask further questions, if you're stuck.
Finally, some advertisement. A great book on intersection theory of plane curves (and elementary algebraic geometry in general) is Fulton's "Algebraic Curves", which is freely available on his webpage. The relevant section on intersection theory of affine plane curves is Chapter 3. Good luck!
