On the separatrix graph of a rational vector field on the Riemann sphere
Kealey Dias, Antonio Garijo

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 where is given by the quotient of two polynomials without common factors on the Riemann sphere. The separatrix graph 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.
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