Darboux integrability and the inverse integrating factor

We mainly study polynomial differential systems of the form dx/dt=P(x,y), dy/dt=Q(x,y), where P and Q are complex polynomials in the dependent complex variables x and y, and the independent variable t is either real or complex. We assume that the polynomials P and Q are relatively prime and that the differential system has a Darboux first integral of the form where the polynomials fi and gj are irreducible, the polynomials gj and hj are coprime, and the λi and μj are complex numbers, when i=1,…,p and j=1,…,q. Prelle and Singer proved that these systems have a rational integrating factor. We improve this result as follows. Assume that H is a rational function which is not polynomial. Following to Poincaré we define the critical remarkable values of H. Then, we prove that the system has a polynomial inverse integrating factor if and only if H has at most two critical remarkable values. Under some assumptions over the Darboux first integral H we show, first that the system has a polynomial inverse integrating factor; and secondly that if the degree of the system is m, the homogeneous part of highest degree of H is a multi-valued function, and the functions exp(hj/gj) are exponential factors for j=1,…,q, then the system has a polynomial inverse integrating factor of degree m+1. We also present versions of these results for real polynomial differential systems. Finally, we apply these results to real polynomial differential systems having a Darboux first integral and limit cycles or foci.