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

摘要

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)(δ)_X + B(X, V)(δ)_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 Computacion of an exponential number of reducible Darboux polynomials. Here we show that the Lagutinskii-Pereira 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.