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 H= f(1)(lambda1) ... f(P)(lambdap) (exp(h(1)/g(1)(n1)))mu(1) ... (exp(h(q)/g(q)(nq)))(muq), where the polynomials f(i) and g(j) are irreducible, the polynomials gj and h(j) are coprime, and the lambda(i) and mu(j) 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 Poincare 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(h(j)/g(j)) 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. (C) 2003 Elsevier Inc. All rights reserved.