2024/07/12 by Leonardo Câmara, Câmara, Leonardo M., Fernando Reis +3 · 1 citation
Mathematics · #(primary) 32M25 #(secondary) 58K45 #32S65 #Algebraic Geometry (math.AG) #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2407.09306
openalex publication_date 2024/07/12 · openalex created_date 2024/07/16 · openalex updated_date 2026/07/28
In this paper, we address one of the most basic and fundamental problems in the theory of foliations and ODEs, the topological invariance of the algebraic multiplicity of a holomorphic foliation. For instance, we prove an adapted version of A'Campo-Lê's Theorem for foliations, i.e., the algebraic multiplicity equal to one is a topological invariant in dimension two. This result is further generalized to higher dimensions under mild conditions; as a consequence, we prove that saddle-nodes are topologically invariant. We prove that the algebraic multiplicity is a topological invariant in several classes of foliations that contain, for instance, the generalized curves and the foliations of second type. Additionally, we address a fundamental result by Rosas-Bazan, which states that the existence of a homeomorphism extending through a neighborhood of the exceptional divisor of the first blow-up implies the topological invariance of the algebraic multiplicity. We show that the result holds if the homeomorphism extends locally near a singularity, even if it does not extend over the entire divisor.