Proximity inequalities and bounds for the degree of invariant curves by foliations of $\mathbb P^2_{\mathbb C}$

In this paper we prove that if $C$ is a reduced curve which is invariant by a foliation $\mathcal F$ in the complex projective plane then one has $\partial ^{\underline {\circ }} C\leq \partial^{\underline {\circ }} \mathcal F+2+a$ where $a$ is an integer obtained from a concrete problem of imposing singularities to projective plane curves. If $\mathcal F$ is nondicritical or if $C$ has only nodes as singularities, then one gets $a=0$ and we recover known bounds. We also prove proximity formulae for foliations and we use these formulae to give relations between local invariants of the curve and the foliation.