Linearization of complex hyperbolic Dulac germs

TL;DR

This paper proves that hyperbolic Dulac germs with complex coefficients can be linearized within a standard quadratic domain, and the linearizing coordinate retains the Dulac germ structure, using hyperbolic transseries normal forms.

Contribution

It establishes the linearizability of complex hyperbolic Dulac germs and shows the linearizing coordinate is also a Dulac germ, extending previous results to complex coefficients.

Findings

Hyperbolic Dulac germs are linearizable on a quadratic domain.

The linearizing coordinate is a complex Dulac germ.

The proof leverages hyperbolic transseries normal forms.

Abstract

We prove that a hyperbolic Dulac germ with complex coefficients in its expansion is linearizable on a standard quadratic domain and that the linearizing coordinate is again a complex Dulac germ. The proof uses results about normal forms of hyperbolic transseries from another work of the authors.