Generic reversible complex polynomial vector fields

TL;DR

This paper classifies and parametrizes generic complex polynomial vector fields of a specific form, describing their bifurcations and providing a realization theorem for their moduli.

Contribution

It introduces a complete parametrization of the moduli space of these vector fields and describes their bifurcation diagrams for degrees 3 and 4.

Findings

Number of generic strata determined

Complete parametrization of strata provided

Bifurcation diagrams described for degrees 3 and 4

Abstract

The paper studies the generic complex 1-dimensional polynomial vector fields of the form $iP(z)\frac{\partial}{\partial z}$, where $P$ is a polynomial 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 the strata is given in terms of a modulus formed by a combinatorial part and an analytic part. The bifurcation diagram is described for degrees 3 and 4. A realization theorem is proved for any generic modulus.