On first integrals admitting asymptotic expansion for holomorphic foliations in dimension two

TL;DR

This paper investigates the dynamics of holomorphic vector fields near singular points in two dimensions, demonstrating the existence of holomorphic first integrals under certain asymptotic expansion conditions.

Contribution

It introduces conditions under which holomorphic first integrals exist for vector fields with asymptotic expansions, advancing understanding of local foliation behavior.

Findings

Existence of holomorphic first integrals under specified hypotheses

Use of asymptotic expansion techniques in local foliation analysis

Conditions for first integrals in small sectors with nonzero asymptotic expansion

Abstract

In this paper we study the dynamics of a holomorphic vector field near a singular point in dimension two using asymptotic expansion techniques. We consider a holomorphic vector field admitting first integrals in small sectors with nonzero asymptotic expansion. Under some natural hypotheses we prove the existence of a holomorphic first integral.