On Brian\c{c}on-Skoda theorem for foliations

TL;DR

This paper extends the Briançon-Skoda theorem to second type foliations, linking Milnor and Tjurina numbers, and establishes bounds for algebraic curves' invariants.

Contribution

It generalizes the Briançon-Skoda theorem for a broader class of foliations and relates key invariants, providing new bounds for algebraic curves.

Findings

Established a relationship between Milnor and Tjurina numbers for second type foliations.

Generalized Mattei's result to a wider class of foliations.

Determined a lower bound for the global Tjurina number of algebraic curves.

Abstract

We generalize Mattei's result relative to the Brian\c{c}on-Skoda theorem for foliations to the family of foliations of the second type. We use this generalization to establish relationships between the Milnor and Tjurina numbers of foliations of second type, inspired by the results obtained by Liu for complex hypersurfaces and we determine a lower bound for the global Tjurina number of an algebraic curve.