Study on Algorithms for D-Modules.

1.与科比大学的Nobuki Takayama教授合作，我定义了Weyl代数的均匀化环上的模的最小（过滤）自由归结的概念（即，具有多项式系数的微分算子的环）。我们还介绍了一个计算均匀化Weyl代数上模的最小自由分解的有效算法。另一方面，利用解析微分算子环关于阶滤子的均匀化，我与M.法国昂热大学的格兰杰说。我们证明了这样一个极小滤自由分解在复形同构下是唯一存在的。作为应用，我们引入了解析超曲面奇异性的一组数值不变量. M.格兰杰和我在由多项式系数算子生成的解析微分算子的齐化环上的有限自由模中找到了一个除法算法。这是T.莫拉为幂级数。Takayama在Nan中实现了该算法。这是D-模的最一般的除法算法之一，它可以处理任何与模结构相容的单项序。N. Takayama，Y. Shiraki和我用D-模的算法研究了特殊参数函数的数值积分方法：通过一些例子，我们证明了这种新方法比经典的数值积分方法更有效。

1. Collaborating with Professor Nobuki Takayama of Kobe University, I defined the notion of minimal (filtered) free resolution for a module over the homogenized ring of the Weyl algebra (i.e., the ring of differential operators with polynomial coefficients). We also introduced an efficient algorithm for computing a minimal free resolution of a module over the homogenized Weyl algebra. This algorithm was implemented in software Nan.On the other -hand, using the homogenization of the ring of analytic differential operators with respect to the order filtration, I defined the notion of minimal filtered free resolution for a module over this homogenized ring of analytic differential operators with Professor M. Granger of Angers University, France. We proved that such a minimal filtered free resolution exists uniquely up to isomorphism of complexes. As an application, we introduced a set of numerical invariants of analytic hypersurface singularities.2. M. Granger and I found a division algorithm in a finite free module over the homogenized ring of analytic differential operators which is generated by operators with polynomial coefficients. This is an extension to D-modules of a celebrated tangent cone algorithm of T. Mora for power series. Takayama implemented this algorithm in Nan. Being able to work with' any monomial ordering compatible with the module-structure, this is one of the most general division algorithms for D-modules.3. N. Takayama, Y. Shiraki and I studied the method of numerical integration for special functions with parameters by using algorithm for D-modules : We showed that for some examples, this, new method is more efficient than the classical 'method of numerical integration.