ON THE NONEXISTENCE OF CONSTANTS OF DERIVATIONS - THE PROOF OF A THEOREM OF JOUANOLOU AND ITS DEVELOPMENT

Inspired by the proof of JOUANOLOU'S non-integrability theorem, we des cribe a method for proving the non-existence of non-trivial constants of some derivations in polynomial rings K [x(1), ..., x(n)], where K i s a field of characteristic 0. In some cases, even the non-existence o f non-trivial constants of derivations in quotient field K (x(1), ..., x(n)) can be proved. When K = R or C, this is equivalent to proving t he non-existence of polynomial, respectively rational, first integrals for some systems of polynomial ordinary differential equations. Sever al examples, among which JOUANOLOU'S one, are described in details.