Some properties of planar polynomial systems of even degree (Q1199810)

Let \(\dot x=P(x,y)\), \(\dot y=Q(x,y)\) be a planar polynomial system and let \(n=\max(\deg(P),\deg(Q))\). The authors show that, if \(n\) is even, then this system must have at least one unbounded trajectory. This implies that the system can have no global centers, that is, singular points \(p\in\mathbb{R}^ 2\) such that \(\mathbb{R}^ 2-\{p\}\) is filled by closed nonsingular trajectories. The proof proceeds by first compactifying the plane using the Poincaré sphere and then analyzing the singular points on the equator of this sphere.