I am working in $k[x,y]$ with two polynomials $F,G$ and the ideal $I = (x,y)$.
I define $n$ and $m$ as the multiplicy of $F$ and $G$ at the point $(0,0)$.
Let's denote $\mathcal{O}$ to be the localization of $k[x,y]$ at the point $(0,0)$.
My goal was to establish a link between the number of intersection of the curves defined by $F$ and $G$ at the point $P=(0,0)$, and the multiplicities $m,\, n$: I want to prove that it is $\geq nm$.
The first step was to establish a link between $nm$ and the $k$-dimension of $k[x,y]/(I^{n+m},F,G)$. I have proved that the dimension is $\geq nm$.
What I need to conclude is to establish a link between the ring $k[x,y]/(I^{n+m},F,G)$ and the local quotient $\mathcal{O}/(F,G)$.
I think that our first ring is precisely isomorphic to $\mathcal{O}/(F,G,I^{n+m})$, but my attempt to prove that the natural map is an isomorphism didn't conclude.
I was looking here for an idea of proof of the fact that it is an isomorphism, or more weakly that $k[x,y]/(I^{n+m},F,G)$ appears as a quotient of $\mathcal{O}/(F,G)$ with more general arguments.
Thank you for your time.
If $R$ is a commutative ring, $\mathfrak m⊂R$ a maximal ideal, and $a,b∈\mathfrak m$, then the factor ring $R/(a,b,\mathfrak m^t)$ is a local ring, its only maximal ideal being $\mathfrak m/(a,b,\mathfrak m^t)$. The localization of $R/(a,b,\mathfrak m^t)$ at $\mathfrak m/(a,b,\mathfrak m^t)$ is $R_{\mathfrak m}/(a,b,\mathfrak m^t)$. But the ring being local its localization at the maximal ideal is the ring itself.
So, your claim "our first ring is precisely isomorphic to $\mathcal O/(F,G,I^{n+m})$" is correct.