Algebraic analysis of residue theory in several complex variables and algorithms

几种复杂变量和算法中留数理论的代数分析

• 批准号: 12640161
• 负责人: TAJIMA Shinichi
• 金额: $ 1.92万
• 依托单位: NIIGATA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640161/
• 关键词: Grothendieck local residue; algebraic local cohomology; D-modules; holonomic system; Grobner basis; isolated singularity; Milnor number; Tjurina number; Grothendieck留数; 代数解析的局所コホモロジー; グレグナ基底

Upon using the theory of holonomic D-modules and methods of computer algebra, we investigated algorithmic aspects of Grothendieck local residues. As applications, we derived and implemented diverse variety of algorithms.1. Algorithm for computing annihilating ideal in the Weyl algebra of a zero dimensional algebraic local cohomology are derived. Main ingredient of this derivation is the notion of holonomic D-modules.2. An algorithm that compute Grothendieck local residues is constructed. The resulting algorithm is efficient and available in use for generic case.3. Some improvement of the above algorithm are also studied.4. Solvability condition for an ordinary differential equation in a space of convergent power series and in a space of formal power series are investigated in the context of algebraic analysis. A necessary and sufficient condition for the solvability is described in terms of local residues. A regular singular system of ordinary differential equation which characterizes algebraic local cohomology solutions of the formal adjoint equation is introduced. The use of this system provides an effective method for computing formal solvability conditions.5. Algebraic local cohomology classes attached to a non quasi homogeneous isolated singularity are studies in the context of D-modules.

利用完整D-模理论和计算机代数方法，研究了Grothendieck局部剩余的算法问题。作为应用，我们推导并实现了多种算法.给出了零维代数局部上同调的Weyl代数中零化理想的计算方法。该推导的主要成分是完整D-模的概念。2.构造了一个计算Grothendieck局部剩余的算法。该算法是有效的，可用于一般情况.并对上述算法进行了改进.利用代数分析方法研究了收敛幂级数空间和形式幂级数空间中常微分方程的可解性条件。一个必要和充分条件的可解性描述的局部残基。引入了一个正则奇异常微分方程组，它刻画了形式伴随方程的代数局部上同调解。该系统的使用为形式可解性条件的计算提供了一种有效的方法.在D-模的背景下研究了非拟齐次孤立奇点的代数局部上同调类。

项目成果

S.Tajima: "非同次常微分方程式の可解条件について"京都大学数理解析研究所講究録. 1168. 66-79 (2000)

S. Tajima：“论非齐次常微分方程的可解性条件”京都大学数学科学研究所 Kokyuroku. 1168. 66-79 (2000)。

S.Tajima, Y.Nakamura: "Unimodal例外型特異点における代数的局所コホモロジー類"京都大学数理解析研究所講究録. 1211. 155-165 (2001)

S. Tajima、Y. Nakamura：“单峰异常奇点的代数局部上同调类”京都大学数学科学研究所 Kokyuroku 1211. 155-165 (2001)。

S.Tajima, Y.Nakamura: "An algorithm for the local residue with the viewpoint of D-modules"Proceedings of the second Congress of International Society for Analysis, its Applications and Computation congress, Kluwer. 809-817 (2000)

S.Tajima、Y.Nakamura：“从 D 模块的角度考虑局部残差的算法”国际分析学会第二届大会、其应用和计算大会的会议记录，Kluwer。

S.Tajima and Y.Nakamura: "Computing point residues for a shape bases case via differential operators"京都大学数理解析研究所講究録. 1158. 87-97 (2000)

S. Tajima 和 Y. Nakamura：“通过微分算子计算形状基情况的点留数”京都大学数学科学研究所 Kokyuroku 1158. 87-97 (2000)。

田島慎一: "非同次常微分方程式の可解条件について"京都大学数理解析研究所講究録. 1168. 66-79 (2000)

Shinichi Tajima：“论非齐次常微分方程的可解性条件”京都大学数学分析研究所 Kokyuroku. 1168. 66-79 (2000)。