Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

In this paper we study planar polynomial differential systems of this form: dXdt=Ẋ=A(X,Y),dYdt=Ẏ=B(X,Y), where A,B∈Z[X,Y]A,B∈Z[X,Y] and degA≤ddegA≤d, degB≤ddegB≤d, ‖A‖∞≤H‖A‖∞≤H and‖B‖∞≤H‖B‖∞≤H. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: D=A(X,Y)∂X+B(X,Y)∂YD=A(X,Y)∂X+B(X,Y)∂Y. Darboux polynomials are usually computed with the method of undetermined coefficients. With this method we have to solve a polynomial system. We show that this approach can give rise to the computation of an exponential number of reducible Darboux polynomials. Here we show that the Lagutinskii–Pereira algorithm computes irreducible Darboux polynomials with degree smaller than NN, with a polynomial number, relatively to dd, log(H)log(H) and NN, binary operations. We also give a polynomial-time method to compute, if it exists, a rational first integral with bounded degree.