function $\xi$ such that $\xi(x,0)=x$. It follows that $$\Phi_{F,G}(x,y)= \left(x,1-\dfrac{x(1-h(\xi(x,y))}{\xi (x,y))}\right)= \left(x,\dfrac{u(x,y)}{\xi(x,y)}\right)$$

The expression defines a meromorphic map in a neighborhood of $(0,0)$. There are two possible cases:

-   $\bf A_1$: the germs $x$ and $\xi$ are relatively prime; the line of poles of $\Phi_{F,G}(x,y)$ is $\xi (x,y)=0$ and has multiplicity 1. We write $\xi(x,y)-x=y\,A_1(x,y)$ for some holomorphic function $A_1(x,y)$; the $\mathcal G$-fiber may be transversal to the $\mathcal F$-fiber (when $A_1(0,0)\neq 0$) or tangent to it (in which case $A_1(0,0)\neq 0$).

-   $\bf A_2$: the germs $x$ and $\xi$ have a common factor; write $\xi(x,y)=x(1+y\,A_2(x,y))$, thus $\Phi_{F,G}(x,y)$ is a holomorphic map ($\mathcal F$ and $\mathcal G$ have $x=0$ as a common fiber), but it may be non-injective (unless $A_2(0,0)\neq 0$).

[**Case B:**]{} We assume:

-   $p=(0,0)$, $C$ is $y=0$ and $\mathcal F$ is defined by $dx=0$;

-   $\phi(p)=(0,0)$, $\mathcal F_0$ is defined by $dX=0$ and $\mathcal G_0$ is defined by $dY-dX=0$ (in affine coordinates);

-   $C_0$ is defined by $Y=h(X)$ with $h(0)=0$, $h^{\prime}(0)=0$ and $\phi(x)= (x,h(x))$.

We have then $$\Phi_{F,G}(x,y)= (x,\xi(x,y)-x +h(\xi(x,y))$$ It follows that $\Phi_{F,G}$ is a holomorphic map in a neighborhood of $p$; writing $\xi(x,y)-x=y\,B(x,y)$, we see that $\Phi_{F,G}$ is a local biholomorphism when $B(0,0)\neq 0$, that 