Intersection multiplicities of Noetherian functions

Consider a foliation defined by two commuting polynomial vector fields V ₁, V ₂ in Cn, and p a non-singular point of the foliation. Denote by L the leaf passing through p, and let F,G∈C[X] be two polynomials. Assume that F{pipe}L=0,G{pipe}L=0 have several common branches. We provide an effective procedure which produces an upper bound for the multiplicity of intersection of remaining branches of F{pipe}L=0 with G{pipe}L=0 in terms of the dimension n and the degrees of V ₁, V ₂, F, G.