Counterexample to a conjecture on the algebraic limit cycles of polynomial vector fields

In Geometriae Dedicata 79 (2000), 101-108, Rudolf Winkel conjectured: for a given algebraic curve f=0 of degree m ≥ 4 there is in general no polynomial vector field of degree less than 2m -1 leaving invariant f=0 and having exactly the ovals of f=0 as limit cycles. Here we show that this conjecture is not true. © Springer 2005.