On a Mattei-Salem theorem

TL;DR

This paper extends the Mattei-Salem theorem by exploring the relationship between valuations of singular foliations and their separatrices, providing new inequalities involving valuation, tangency excess, and degree.

Contribution

It generalizes the Mattei-Salem theorem to broader conditions and establishes inequalities linking valuation, tangency excess, and degree of holomorphic foliations.

Findings

Extended the Mattei-Salem theorem to new conditions

Derived inequalities relating valuation, tangency excess, and degree

Provided insights into the structure of singular foliations on complex surfaces

Abstract

We investigate the relationship between the valuations of a germ of a singular foliation $\mathcal{F}$ on the complex plane and those of a balanced equation of separatrices for $\mathcal{F}$, extending a theorem by Mattei-Salem. Under certain conditions, we also derive inequalities involving the valuation, tangency excess, and degree of a holomorphic foliation $\mathcal{F}$ on the complex projective plane.