Generalized rational first integrals of analytic differential systems

In this paper we mainly study the necessary conditions for the existence of functionally independent generalized rational first integrals of ordinary differential systems via the resonances. The main results extend some of the previous related ones, for instance the classical Poincaré?s one (Poincaré, 1891, 1897 16]), the Furta?s one (Furta, 1996 8]), part of Chen et al.?s ones (Chen et al., 2008 4]), and the Shi?s one (Shi, 2007 18]). The key point in the proof of our main results is that functionally independence of generalized rational functions implies the functionally independence of their lowest order rational homogeneous terms.