Generic reversible complex polynomial vector fields

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.

arXiv comment20 pages, 11 figures, one table