I am reading kenji uneoگس book on algebraic geometry 1. I don't understand how to compute the local intersection multiplicity. I would appreciate if you can show me how to compute it for the next two curves. $f=y-2x ,g=y^2-x^2(x+1)$.
Thanks in advance.
Edit: I've searched the net, and found a nice coverage of how to calculate the local intersection multiplicity, here: http://www.maths.manchester.ac.uk/~gm/teaching/MATH32062/intmult.pdf
But I still don't see from this how to calculate for example the local intersection multiplicity at the origin of the two curves:
$$f= y^2-x^3 ; g= y^2-x^2(x+1)$$
There seems to be only an algorithm for when one equation is linear and the other isn't linear.
The link you provided missed out two properties of the intersection number, which are all I've ever used to calculate intersection numbers!
1) $I_P(F,G)\geq m_P(F)m_P(G)$ with equality if $F,G$ have no common tangent. $m_P$ is the multiplicity of the curve at $P$.
2) $I_P(F,G) = I_P(F,G+AF),\ A\in k[X,Y]$
Creative use of these two rules usually works. Use 2) until you have no common tangents, then apply 1).
In your case, $F=Y^2-X^3$ and $G=Y^2-X^2(X+1)$ The least degree homogeneous components are $F_2 = Y^2$ and $G_2=Y^2-X^2=(Y-X)(X+Y)$ so $F,G$ have no common tangents at the origin, and therefore $I_P(F,G) = m_P(F)m_P(G) = 2\cdot 2 = 4$.