Study on Algorithms for D-Modules.

D-模块算法研究。

• 批准号: 13640192
• 负责人: OAKU Toshinori
• 金额: $ 2.5万
• 依托单位: Tokyo Woman's Christian University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640192/
• 关键词: D-module; lihear partial differential equation; algorithm; Groebner base; free resolution; minimal resolution; symbolic computation; division; 線形編微分方程式; 線形微分方程式; 微分作用素; Groebner基底; 微分作用素環

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.

1. 与神户大学的 Nobuki Takayama 教授合作，我定义了 Weyl 代数均质环（即具有多项式系数的微分算子环）上的模的最小（过滤）自由分辨率的概念。我们还引入了一种有效的算法，用于在均质化韦尔代数上计算模块的最小自由分辨率。该算法是在软件 Nan 中实现的。另一方面，利用解析微分算子环相对于阶次过滤的均质化，我定义了在该均质解析微分环上的模块的最小过滤自由分辨率的概念与法国昂热大学 M. Granger 教授合作。我们证明了这种最小的过滤自由分辨率在配合物的同构上是唯一存在的。作为一个应用，我们引入了一组解析超曲面奇点的数值不变量。2． M. Granger 和我在解析微分算子的均质环上的有限自由模块中发现了一种除法算法，该环由具有多项式系数的算子生成。这是 T. Mora 著名的幂级数切锥算法的 D 模块的扩展。 Takayama 在 Nan 中实现了这个算法。能够使用与模块结构兼容的任何单项式排序，这是 D 模块最通用的除法算法之一。3。 N. Takayama、Y. Shiraki 和我通过使用 D 模算法研究了带参数的特殊函数的数值积分方法：我们表明，对于一些例子，这种新方法比经典的“数值积分方法”更有效。