We study the relationship between the Milnor and Tjurina numbers of a singular foliation \(\mathcal {F}\) , in the complex plane, with respect to a balanced divisor of separatrices \(\mathcal B\) for \(\mathcal {F}\) . For that, we associate with \(\mathcal {F}\) a new number called the \(\chi \) -number and we prove that it is a \(C^{1}\) invariant for holomorphic foliations. We compute the polar excess number of \(\mathcal {F}\) with respect to a balanced divisor of separatrices \(\mathcal {B}\) for \(\mathcal {F}\) , via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of \(\mathcal {B}\) and the \(\chi \) -number of \(\mathcal {F}\) . On the other hand, we generalize, in the plane case and the formal context, the well-known result of Gómez-Mont given in the holomorphic context, which establishes the equality between the GSV-index of the foliation and the difference between the Tjurina number of the foliation and the Tjurina number of a set of separatrices of \(\mathcal {F}\) . Finally, we state numerical relationships between some classic indices, as Baum–Bott, Camacho–Sad, and variational indices of a singular foliation and its Milnor and Tjurina numbers; and we obtain a bound for the sum of Milnor numbers of the local separatrices of a holomorphic foliation on the complex projective plane.
