On the separatrix graph of a rational vector field on the Riemann sphere

TL;DR

This paper characterizes the structure of the separatrix graph of rational vector fields on the Riemann sphere, providing a topological description of the boundary between different trajectory types.

Contribution

It offers a novel characterization of the separatrix graph's properties, linking planar directed graphs to rational vector fields on the Riemann sphere.

Findings

Characterization of the separatrix graph's properties

Topological conditions for a graph to be a separatrix graph

Insights into the boundary structure of rational flows

Abstract

We consider the rational flow $\xi_R(z)= R(z) (d/dz)$ where $R$ is given by the quotient of two polynomials without common factors on the Riemann sphere. The separatrix graph $\Gamma_R$ is the boundary between trajectories with different properties. We characterize the properties of a planar directed graph to be homeomorphic to the separatrix graph of a rational vector field on the Riemann sphere.