Remarks on rational vector fields on $\mathbb C\mathbb P^1$

TL;DR

This paper introduces geometric tools like periodgons, star domains, and translation surfaces to analyze bifurcations in families of rational vector fields on the complex projective line, especially focusing on degree 4.

Contribution

It generalizes existing geometric objects to rational vector fields and applies these tools to study bifurcations in degree 4 cases.

Findings

Introduction of geometric objects for rational vector fields

Description of bifurcations using these objects

Specialization to degree 4 vector fields

Abstract

In this paper we introduce geometric tools to study the families of rational vector fields of a given degree over $\mathbb C\mathbb P^1$. To a generic vector field of such a parametric family we associate several geometric objects: a periodgon, a star domain and a translation surface. These objects generalize objects with the same name introduced in previous works on polynomial vector fields. They are used to describe the bifurcations inside the families. We specialize to the case of rational vector fields of degree $4$.