In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.

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Abstract. We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach is inspired by an idea of Ferragut and Giacomini ([FG10]). We improve upon their work by proving that rational first integrals can be computed via systems of linear equations instead of systems of quadratic equations. The main ingredients of our algorithms are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating a power series. This leads to a probabilistic algorithm with arithmetic complexityÕ(N 2ω ) and to a deterministic algorithm solving the problem iñ O(d 2 N 2ω+1 ) arithmetic operations, where N denotes the given bound for the degree of the rational first integral, and where d ≤ N is the degree of the vector field, and ω the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, inÕ(N ω+2 ) arithmetic operations. By comparison, the best previous algorithm given in [Chè11] uses at least d ω+1 N 4ω+4 arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package RationalFirstIntegrals which is available to interested readers with examples showing its efficiency.

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

A recent algorithmic procedure for computing the absolute factorization of a polynomial P (X, Y ), after a linear change of coordinates, is via a factorization modulo X 3 . This was proposed by A. Galligo and D. Rupprecht in [16], [8]. Then absolute factorization is reduced to finding the minimal zero sum relations between a set of approximated numbers bi, i = 1 to n such that P n i=1 bi = 0, (see also [17]). Here this problem with an a priori exponential complexity, is efficiently solved for large degrees (n > 100). We rely on L.L.L. algorithm, used with a strategy of computation inspired by van Hoeij's treatment in [23]. For that purpose we prove a theorem on bounded integer relations between the numbers bi, also called linear traces in [19]

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Let f (X, Y ) ∈ Z[X, Y ] be an irreducible polynomial over Q. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of f , or more precisely, of f modulo some prime integer p. The same idea of choosing a p satisfying some prescribed properties together with LLL is used to provide a new strategy for absolute factorization of f (X, Y ). We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to factorize some polynomials of degree up to 400.

In this article we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is inÕ(N ω+1 ), where N is the bound on the degree of a representation of the first integral and ω ∈ [2; 3] is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on authors' websites. In the last section, we give some examples showing the efficiency of these algorithms. First integrals and Symbolic computations and Complexity analysis 1