Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
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DOI：
10.1016/j.aam.2014.03.001
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发表时间：
2014-01
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影响因子：
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通讯作者：
Carlos E. Arreche
中科院分区：
文献类型：
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作者：
Carlos E. Arreche
本文提出了一种新的方法来计算微分伽罗瓦群H的幂幺根Ru（H），它与一个形式为<$2 <$x2 Y− q Y= 0的参数化二阶齐次线性微分方程相关联，其中q∈ F（x）是x中的有理函数，其系数在特征为零的域F中，并且<$x是参数导子的交换集. Dreyfus提出的方法将Ru（H）的计算简化为求解一个创造性的伸缩问题，该问题的有效解依赖于最大约简商H/Ru（H）是n-常数线性微分代数群的假设.当这个条件不满足时，可以有效地计算一组新的参数导子H ′，使得相关的微分伽罗瓦群H′具有H′/Ru（H′）是H ′-常数的性质，并且使得Ru（H）由与Ru（H′）相同的微分方程定义。这样，Ru（H）的计算就归结为Ru（H′）的有效计算.我们预计，这种方法的阐述将是成功的，在扩展的适用性最近开发的一些算法Minchenko，Ovchinnikov和辛格计算幂单根高阶方程。
We propose a new method to compute the unipotent radical R u (H) of the differential Galois group H associated to a parameterized second-order homogeneous linear differential equation of the form∂ 2∂ x 2 Y− q Y= 0, where q∈ F (x) is a rational function in x with coefficients in a Π-field F of characteristic zero, and Π is a commuting set of parametric derivations. The procedure developed by Dreyfus reduces the computation of R u (H) to solving a creative telescoping problem, whose effective solution depends on the assumption that the maximal reductive quotient H/R u (H) is a Π-constant linear differential algebraic group. When this condition is not satisfied, one can effectively compute a new set of parametric derivations Π′ such that the associated differential Galois group H′ has the property that H′/R u (H′) is Π′-constant, and such that R u (H) is defined by the same differential equations as R u (H′). Thus the computation of R u (H) is reduced to the effective computation of R u (H′). We expect that an elaboration of this method will be successful in extending the applicability of some recent algorithms developed by Minchenko, Ovchinnikov, and Singer to compute unipotent radicals for higher order equations.