We generalize Mattei’s result relative to the Briançon–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.

中文翻译：

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关于叶状结构的布里昂松-斯科达定理

我们将 Mattei 相对于叶状结构的 Briançon-Skoda 定理的结果推广到第二类叶状结构族。受到 Liu 获得的复杂超曲面结果的启发，我们使用这种概括来建立第二类叶状结构的 Milnor 数和 Tjurina 数之间的关系，并确定代数曲线的全局 Tjurina 数的下界。