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We prove that, for generic bounded polynomial vector fields in R" with isolated critical points, the sum of the indices at all their critical points is (-1)" . We characterize the local phase portrait of the isolated critical points at infinity for any bounded polynomial vector field in R" . We apply this characterization to show that there are exactly seventeen different behaviours at infinity for bounded cubic polynomial vector fields in the plane.doi:10.1090/s0002-9947-1990-0998352-5 fatcat:4nk7y6q5abhkpbtldcrvhl5hfi