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

Abstract

International audienceIn this paper we study planar polynomial differential systems of this form: dX/dt=A(X, Y ), dY/dt= B(X, Y ), where A,B belongs to Z[X, Y ], degA ≤ d, degB ≤ d, and the height of A and B is smaller than H. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: D =A(X, Y )dX + B(X, Y )dY . 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

Symbolic Computations of First Integrals for Polynomial Vector Fields

“…These two steps correspond to two practical difficulties. The computation of Darboux polynomials with bounded degree can be performed in polynomial time, see [10]. This method is based on the so-called extactic curve introduced by Pereira in [29] and uses a number of binary operations that is polynomial in the bound N , the degree d and the logarithm of the height of A and B.…”

“…This method is based on the so-called extactic curve introduced by Pereira in [29] and uses a number of binary operations that is polynomial in the bound N , the degree d and the logarithm of the height of A and B. Unfortunately, the arithmetic complexity of this computation is in O(N 4ω+4 ), see [10]. The recombination part can be solved with linear algebra if we are looking for Darbouxian first integrals.…”

On integration of the first order differential equations in a finite terms

“…So in all known algorithms for solution of problem 1 the user must give a boundary for degree of required integral [4,5]. This boundary has simple geometric sense so we can state the following variant of Beaune problem.…”

“…Computation of Lagutinski determinant of a big order is a difficult task [4] but for the solution of the problem 2 it is enough to calculate determinant in one point with random integer coordinates. If…”

Finding non-polynomial positive invariants and lyapunov functions for polynomial systems through Darboux polynomials

“…These methods require running time that is exponential in the size of the ODE, in the worst case. A recent approach based on the "ecstatic curve" was proposed by Chèze [23] reduces the problem of finding Darboux polynomials to factoring a large polynomial that is expressed as a determinant of polynomials. This approach has been shown to be polynomial time provided the coefficients of the ODE and the successive Lie derivatives belong to the field Z that admits efficient factorization.…”