transactions of the american mathematical society Volume 318. Number 2. April 1990 BOUNDED POLYNOMIAL VECTOR FIELDS ANNA CIMA AND JAUME LLIBRE Abstract. 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 char- acterization to show that there are exactly seventeen different behaviours at infinity for bounded cubic polynomial vector fields in the plane. 0. Introduction Let X : U —> Rk be a vector field where U is an open set of Rk . Let y(t) = y(t, x) be the integral curve of X such that y(0) = x. Let Ix be its maximal interval of definition. We shall say that X is a bounded vector field if for all x e U, there exists some compact set K c U such that y(t) G K for each t g Ix n (0, +00). In §1 we introduce the stereographic compactification of X, s(X). We then use the index formula of Bendixson and the Poincaré-Hopf theorem to prove the following result: Proposition A. Let X be a bounded polynomial vector field in the plane. If all the critical points of s(X) are isolated, then the sum of the indices at all those critical points is 1. In §2 we use the Poincaré compactification of X, p(X), to characterize the local phase portrait of the isolated critical points at infinity for bounded poly- nomial vector fields X = (P, Q) in the plane. The degree n of X is defined by n = max{degreeP, degree Q}. We denote by ix(q) the index of Y at a critical point q of X. We then prove the following theorem: Theorem B. Let X be a bounded polynomial vector field in the plane. If q is an isolated infinite critical point of X, then (a) The local phase portrait of p(X) at q is described in Figure 2.2 (resp. Figure 2.4) when the degree of X is even (resp. odd). Received by the editors July 12, 1988. The contents of this paper have been presented to the meeting "Qualitative Theory of Differential Equations" in Szeged, Hungary, August 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 34C05; Secondary 58F14. Key words and phrases. Bounded vector field, index, blow-up. The two authors have been partially supported by a CICYT grant PB 86-0351. This work is a chapter of the Ph.D. thesis of the first author; see [C]. ©1990 American Mathematical Society 0002-9947/90 $1.00+ $.25 per page 557 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use