Study on Algorithms for D-Modules

d -模表示微分算子环上的模，它对应于线性偏微分方程组。d模理论自20世纪70年代以来主要是由佐藤、柏原和川合发展起来的。然而，目前还没有对d模的实际计算进行系统的研究。通过与N.Takayama的合作，我发现了将Groebner基技术应用于d -模的d -模算法。特别是，我们已经找到了计算与D-模的逆象（限制）和正象（积分）相关的上同调群的算法。这些算法的基本组成部分之一是计算适用于过滤（或权重向量）的d模块的自由分辨率。然而，我们获得的自由分辨率往往太大，无法完成计算。为了克服这一瓶颈，我们引入了适用于过滤的最小自由分辨率的概念，并证明了这种最小自由分辨率是可计算的。这种最小分辨率算法使我们能够更有效地计算代数de Rham上同群和代数局部上同群。N.Takayama在他的计算机代数系统Kan中实现了这种算法，并通过互联网向公众开放。

D-modules stand for modules over the ring of differential operators, which corresponds to systems of linear partial differential equations. The theory of D-modules has been developed since 1970's mainly by M.Sato, M.Kashiwara and T.Kawai. However, there was no systematic study on actual computation for D-modules. By the collaboration With N.Takayama, I have found algorithms for D-modules applying Groebner basis techniques to D-modules. In particular, we have found algorithms for computing the cohomology groups associated with the inverse image (restriction) and the direct image (integration) of a D- module. One of the essential ingredients in these algorithsms is the computation of free resolutions for D-modules which are adapted to a filtration (or a weight vector). However free resolutions we obtained were often too big to complete the computation.In order to overcome this bottleneck, we have introduced the notion of minimal free resolution that is adapted to a filtration and have shown that such a minial free resolution is computable. This minimal resolution algorithm enables us to compute, e.g., algebraic de Rham cohomology groups and algebraic local cohomology groups more effciently. N.Takayama has implemented this algorithm in his computer algebra system Kan and has opened it to the public via internet.