Bounded polynomial vector fields

We prove that, for generic bounded polynomial vector fields in Rn with isolated critical points, the sum of the indices at all their critical points is (−1)n. We characterize the local phase portrait of the isolated critical points at infinity for any bounded polynomial vector field in Rn. We apply this characterization to show that there are exactly seventeen different behaviours at infinity for bounded cubic polynomial vector fields in the plane. © 1990 American Mathematical Society.