达布多项式和有理首次积分的多项式时间计算
Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time
摘要：
本文研究如下形式的平面多项式微分系统：dX/dt=A（X，Y），dY/dt= B（X，Y），其中A，B属于Z[X，Y]，degA \leq d，degB \leq d，且A和B的高度小于H.平面多项式微分系统的许多性质都与其导子D =A（X，Y）dX + B（X，Y）dY的不可约达布多项式有关。达布多项式的计算通常采用待定系数待定系数法。用这种方法，我们必须解决一个多项式系统。我们表明，这种方法可以产生的指数数量的可约达布多项式的计算。在这里，我们表明，Lagutinskiii-Pereira的算法计算不可约达布多项式的次数小于N，与多项式的数量，相对于d，log（H）和N，二元运算。如果有理首次积分存在，我们也给出了一个多项式时间的方法来计算它。
英文摘要：
In 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 \leq d, degB \leq 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.
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