Generic Complex Polynomial Vector Fields with Real Coefficients

TL;DR

This paper classifies complex polynomial vector fields with real coefficients, providing a detailed parametrization of their generic structures, analyzing bifurcations for degree 4, and establishing a realization theorem for these moduli.

Contribution

It offers a complete parametrization of generic complex polynomial vector fields with real coefficients and describes bifurcations and realization results.

Findings

Determined the number of generic strata.

Provided a complete parametrization of these strata.

Described the bifurcation diagram for degree 4.

Abstract

The paper studies the complex 1-dimensional polynomial vector fields with real coefficients under topological orbital equivalence preserving the separatrices of the pole at infinity. The number of generic strata is determined, and a complete parametrization of these strata is given in terms of a modulus formed by a combinatorial and an analytic part. The bifurcation diagram is described for the degree 4. A realization theorem is proved for any generic modulus.