On the critical points of planar polynomial Hamiltonian systems

Abstract

It is well known that the critical points of planar polynomial Hamiltonian vector fields are either centers or points with an even number of hyperbolic sectors. We give a sharp upper bound of the number of centers that these systems can have in terms of the degrees of their components. We also prove that generically the critical points at infinity of their Poincaré compactification are either nodes or have indices $-1,0$ or 1 and have at most two sectors: both hyperbolic, both elliptic or one of each type. These characteristics are no more true in the non generic situation. Although these results are known we revisit their proofs. The new proofs are shorter and based on a new approach.