fact, $P(\Phi_{F,G}(x,0))=0$ and $
P(X,Y)=a_{0d}Y^d + \sum_{j\leq d-1}a_{ij}X^iY^j
$ and therefore $$P(\Phi_{F,G}(x,y))= a_{0d}\dfrac{u^d}{\xi^d}+ \dfrac{\sum_{d-j\geq 1}a_{ij}x^iu^j{\xi}^{d-j}}{\xi^d}$$ In particular, $v(x,0)=x^{d-1}A_1(x,0)+\dots$. We have also $L(\Phi_{F,G}(x,y))=x+b$, $b\neq 0$, so that $$\Phi_{F,G}^{*}\,\omega = \dfrac{1}{\xi^{d+1}}[(x+b){\xi}(yd\,v + vd\,y) - d.yv((x+b)d\xi + {\xi}d\,x)]$$ Therefore $\mathcal L^{*}$ is defined by $ (x+b){\xi}(yd\,v + vd\,y) - d.yv((x+b)d\xi + {\xi}d\,x)=0$ near the point $p$ and $$Z(\mathcal L^{*},C,p)= 1+mult_0(v(x,0))= d+mult_0(A_1(x,0))$$ We observe that $Z(\mathcal L^{*},C,p)>0$ when the case ${\bf A1}$ is present.

[**Case A2**]{}: $\xi$ divides $x$ ($\mathcal F$ and $\mathcal G$ share the leaf passing through $p$). Let us write as before $\xi(x,y)= x(1+yA_2(x,y))$; it follows that $$\Phi_{F,G}(x,y)=(x, \dfrac {yA_2(x,y) +h(\xi(x,y))}{1+yA_2(x,y)})$$ Writing $P(\Phi_{F,G}(x,y))= yv(x,y)$, we see that $v(x,0)=A_2(x,0)+\dots$ and $$\Phi_{F,G}^{*}\,\omega = (x+b)(vdy + ydv) - d.yvdx$$ We conclude that $$Z(\mathcal L^{*},C,p)=mult_0(v(x,0))=mult_0(A_2(x,0))$$ Let us notice that $Z(\mathcal L^{*},C,p)=0$ implies that $A_2(0,0)\neq 0$, that is, $\Phi_{F,G}(x,y)$ is a local biholomorphism at $p$.

[**Case B**]{}: $\phi(p)\notin tang(\mathcal F_0,\mathcal G_0) \cap C_0$. We have

$$\Phi_{F,G}(x,y)=(x,\xi-x+h(\xi(x,y))$$ Writing $P(\Phi_{F,G}(x,y))= yv(x,y)$, we see that $v(x,0)=B(x,0)+\dots$ and $$\Phi_{F,G}^{*}\,\omega = (x+b)(vdy + ydv) - d.yvdx$$ We conclude that $$Z(\mathcal L^{*},C,p)=mult_0(v(x,0))_0=mult_0(B(x,0))$$ Again, $Z(\mathcal L^{*},C,p)=0$ implies that $B(0,0)\neq0$, that is, $\Phi_{F,G}(x,y)$ is a local biholomorphism at the point p.

We intend now to see the implications of having two maps $\Phi_{F,G}$ and 