Some new results on Darboux integrable differential systems (Q439279)

The author studies complex planar polynomial differential systems having a Darboux first integral. He defines remarkable values and remarkable curves associated with Darboux first integrals. Remarkable values and remarkable curves were first defined by Poincaré for rational first integrals; their importance was first shown in [\textit{J. Chavarriga, H. Giacomini, J. Giné} and \textit{J. Llibre}, J. Differ. Equations 194, No. 1, 116--139 (2003; Zbl 1043.34001)].NEWLINENEWLINEIn the present paper, these definitions are extended to Darboux first integrals, a result is proved that characterizes the existence of a polynomial inverse integrating factor by means of the number of critical remarkable values for these systems. The infinity is studied, through the so-called characteristic polynomial \(\mathcal{F}=x \tilde{Q}-y \tilde{P}\), where \(\tilde{P}\) and \(\tilde{Q}\) are the homogeneous parts of highest degree of \(P\) and \(Q\), the polynomials that define the differential system, and its relation with the inverse integrating factors of the system. The importance of the numerator of the exponential factor of the Darboux first integral for the construction of \(\mathcal{F}\) is shown.