Note on a paper of J. Llibre and G. Rodríguez concerning algebraic limit cycles

In a recent paper of Llibre and Rodríguez (J. Differential Equations 198 (2004) 374–380) it is proved that every configuration of cycles in the plane is realizable (up to homeomorphism) by a polynomial vector field of degree at most 2(n+r)-1, where n is the number of cycles and r the number of primary cycles (a cycle C is primary if there are no other cycles contained in the bounded region limited by C). In this letter we prove the same theorem by using an easier construction but with a greater polynomial bound (the vector field we construct has degree at most 4n-1). By using the same technique we also construct polynomial vector fields realizing (up to homeomorphism) any configuration of limit cycles which can be linked and knotted in . This answers a question of R. Sverdlove.