Polynomial differential equations with many real ovals in the same algebraic complex solution
Let FolR(2, d) be the space of real algebraic foliations of degree d in RP(2). For fixed d, let IntR(2, d) = {F 2 FolR(2, d) | F has a non-constant rational first integral}. Given F 2 IntR(2, d), with primitive first integral G, set O(F) = number of real ovals of the generic level (G = c). Let O(d) = sup{O(F) | F 2 IntR(2, d)}. The main purpose of this paper is to prove that O(d) = +1 for all d _ 5.
Vector fields, invariant varieties and linear systems
Nous etudions les relations entre les varietes invariantes des champs de vecteurs et le lieu d'inflexion des systemes lineaires par rapport aux champs de vecteurs. Parmi les consequences de cette investigation nous obtenons un critere pour l'existence d'integrales premieres rationnelles d'un degre donne, une borne pour le nombre d'integrales premieres dans les familles de champs vecteurs et une generalisation du critere de Darboux. Nous donnons aussi une nouvelle preuve du resultat de Gomez-Mont sur les feuilletages dont toutes les feuilles sont algebriques.
Some examples for the Poincaré and Painlevé problems
Abstract In 1891, Poincare started a series of three papers in which he tried to answer the following question (cf. [21–23]): “Is it possible to decide if an algebraic differential equation in two variables is algebraically integrable?” (in the sense that it has a rational first integral). More or less at the same time P. Painleve asked the following question: “Is it possible to recognize the genus of the general solution of an algebraic differential equation in two variables which has a rational first integral?”. In this paper we give examples of one-parameter families which show that both problems have a negative answer. With some of the families we can also answer a question posed by M. Brunella in [5].