Intersection multiplicities of Noetherian functions

Consider a foliation defined by two commuting polynomial vector fields V1,V2V1,V2 in CnCn, and pp a non-singular point of the foliation. Denote by LL the leaf passing through pp, and let F,G∈C[X]F,G∈C[X] be two polynomials. Assume that F|L=0,G|L=0F|L=0,G|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|L=0F|L=0 with G|L=0G|L=0 in terms of the dimension nn and the degrees of V1,V2,F,GV1,V2,F,G.