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: dX dt = Ẋ = A(X, Y ), dY dt = Ẏ = B(X, Y ), where A, B ∈ Z[X, Y ] and degA ≤ d, degB ≤ d, ‖A‖∞ ≤ H and ‖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 )∂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’s algorithm computes irreducible Darboux polynomials with degree smaller than N , with a polynomial number, relatively to d, log(H) and N , binary operations. We also give a polynomial-time method to compute, if it exists, a rational first integral with bounded degree. Introduction In this paper we study the following planar polynomial differential system: dX dt = Ẋ = A(X,Y ), dY dt = Ẏ = B(X,Y ), where A,B ∈ Z[X,Y ] and degA ≤ d, degB ≤ d, ‖A‖∞ ≤ H and ‖B‖∞ ≤ H. We associate to this polynomial differential system the polynomial derivation D = A(X,Y )∂X +B(X,Y )∂Y . A polynomial f is said to be a Darboux polynomial, if D(f) = g.f , where g is a polynomial. A lot of properties of a polynomial differential system are related to irreducible Darboux polynomials of the corresponding derivation D. Usually Darboux polynomials are computed with the method of undetermined coefficients. In other words, if we suppose that deg f ≤ N then D(f) = g.f gives a polynomial system in the unknown coefficients of g and f . Then we can find f and g if we solve this system. We will see that this strategy can give rise to the computation of an exponential number of reducible Darboux polynomials. In this paper we show that we can compute all the irreducible Darboux polynomials of degree smaller than N with O (