Analytic normal forms for planar resonant saddle vector fields

TL;DR

This paper establishes essentially unique normal forms for holomorphic vector fields near a p:q resonant saddle singularity, providing explicit polynomial models that serve as local analytic conjugates.

Contribution

It introduces explicit, essentially unique normal forms for resonant saddle vector fields, solving a longstanding problem in local analytic classification.

Findings

Normal forms are essentially unique.

Explicit polynomial models are provided.

Addresses a long-standing classification problem.

Abstract

We give essentially unique ``normal forms'' for germs of a holomorphic vector field of the complex plane in the neighborhood of an isolated singularity which is a p:q resonant-saddle. Hence each vector field of that type is conjugate, by a germ of a biholomorphic map at the singularity, to a preferred element of an explicit family of vector fields. These model vector fields are polynomial in the resonant monomial.Abstract. This work is a followup of a similar result obtained for parabolic diffeomorphisms which are tangent to the identity, and solves the long standing problem of finding explicit local analytic models for resonant saddle vector fields.