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 = X = A(X, Y ), dY dt = Y = B(X, Y ), where A, B Z[X, Y ] and deg A d, deg B 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.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Guillaume Chèze
• Keywords: CORR 2010, Darboux Polynomials, Education, Planar Polynomial, Polynomial