Around Jouanolou non-integrability theorem
The authors extend the Jouanolou nonintegrability theorem on the first-order polynomial ODEs \(x_1'=x_2^s\), \(x_2'=x_3^s, \dots,x_{n-1}' =x_n^s\), \(x_n'=x_1^s\), \(s\in \mathbb{N}\), \(s\geq 2\), to \(n\geq 3\), \(s\geq 3\). Putting the system in the form of a derivation on the polynomial ring \(C[x_1,x_2, \dots,x_n]\), they elegantly show in an elementary way that this derivation does not allow a Darboux polynomial for \(n\geq 5\) and arbitrary \(s\geq 3\), or for \(n=3\) and arbitrary \(s\geq 5\). In a separate section it is proved that also in the case \(n=3\) and \(s=3,4\), the above derivation has no polynomial constant. In the last section, the authors study the generic nonintegrability of homogeneous polynomial derivations.
scientific article; zbMATH DE number 1611984