Equações de Pfaff e a não existência de soluções algébricas

Abstract

In 1979, J.P.Jouanolou, in his book ”Equations de Pfaff Algébriques”[12], presents a density’s result which says that the set of Pfaff’s algebraic equations of degree m > 2 in P2 without algebraic solutions is dense in the set of Pfaff’s algebraic equations. As this is a result about density, it is necessary to ensure that the set of Pfaff’s algebraic equations without algebraic solutions is not empty. In order to do it, Jouanolou presents in the same paper an example of Pfaff’s equation without algebraic solution. In this work, we study the example of Jouanolou, based on the Zoladek’s article [23]. The author brings a more analytical approach to this problem and presents one proof based on a generalization of the Integration Theorem of Darboux (see [4]) proposed by the author in the same article.