Analytic normal forms for convergent saddle-node vector fields

TL;DR

This paper develops unique analytic normal forms for saddle-node vector fields in the complex plane, enabling explicit computation near singularities with convergent separatrices.

Contribution

It introduces a method to explicitly compute analytic normal forms for saddle-node vector fields with convergent separatrices, advancing the understanding of their local structure.

Findings

Provides explicit formulas for normal forms

Ensures uniqueness of the analytic normal forms

Facilitates computation near saddle-node singularities

Abstract

We give unique analytic "normal forms" for germs of a holomorphic vector field of the complex plane in the neighborhood of an isolated singularity of saddle-node type having a convergent formal separatrix. We specifically address the problem of computing the normal form explicitly.