intersection multiplicity at non-zero point

Question

Compute the intersection multiplicity of $f=x+y-2$ and $g=x^2+y^2-2$ at $(1,1)$. Is this the same as the intersection multiplicity of $f(x+1)$ and $g(x+1)$ at $(0,0)$ which I have computed to be 2?

Thanks

Answer 1

You can apply the automorphism $(x,y)\mapsto (x+1,y+1)$ and the intersection is the same as the intersection of $f(x+1,y+1)=x+y$ with $g(x+1,y+1)=x^2+2x+y^2+2y$ at $(0,0)$, which is $2$ as you said: replacing the parametrisation $(t,-t)$ of the first in the second you get $2t^2$ so the intersection number is $2$.

Answer 2

In section 3 of chapter 3 of William Fulton's book we find this:

If $T$ is an affine change of coordinates on $\mathbb{A}^2$, and $T(Q)=P$, then $I(P,F\cap G)=I(Q,F^T\cap G^T)$, where $I(P,F\cap G)$ stands for the intersection number of the curves $F$ and $G$ at $P$. So apparently you're right.

You can download the book for free at this link http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf