The dynamics of the Jouanolou foliation on the complex projective 2-space (Q2730709)

The natural extension to \(\mathbb{C}\mathbb{P}(2)\) of the differential equation \(P(x,y)dy-Q(x,y)dx=0\) where \(P\) and \(Q\) are polynomials with complex coefficients defines a complex one-dimensional foliation \({\mathcal F}\) on \(\mathbb{C}\mathbb{P}(2)\). A common zero of \(P\) and \(Q\) is said to be a singular point of \({\mathcal F}\). Assume that all singular points of \({\mathcal F}\) are isolated. A minimal set of \({\mathcal F}\) is an invariant closed non-empty subset of \(\mathbb{C}\mathbb{P}(2)\) which is minimal with these three properties. A minimal set is non-trivial if it is not a singular point. The paper studies an example due to Jouanolou of a foliation in which the degrees of \(P\) and \(Q\) are \(\leq 5\) for which the authors prove that it has no non-trivial minimal set. The proof uses reliable computations based on interval arithmetic (AWA program).