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Study on Algorithms for D-Modules

Research Project

Project/Area Number 09640055
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionYokohama City University

Principal Investigator

OAKU Toshonori  Yokohama City Univ., Dept. of Mathematical Sciences, Associate Professor, 理学部, 助教授 (60152039)

Co-Investigator(Kenkyū-buntansha) FUJII Michihiko  Yokohama City Univ., Dept. of Mathematical Sciences, Assistant, 理学部, 助手 (60254231)
FUJII Kazuyuki  Yokohama City Univ., Dept. of Mathematical Sciences, Associate Professor, 理学部, 助教授 (00128084)
ASANO Hiroshi  Yokohama City Univ., Dept. of Mathematical Sciences, Professor, 理学部, 教授 (00046012)
TAKAYAMA Nobuki  Kobe Univ., Dept. of Mathematics, Professor, 理学部, 教授 (30188099)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 1998: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1997: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsD-module / linear partial differential equation / algorithm / Groebner base / symbolic computation
Research Abstract

Our purpose has been to establish algorithms for computing various operations (functors) for D-modules by applying the method of Groebner base to the ring of differential operators and to develop software for doing such computations. Our results are as follows :
1. Algorithms for computing functors of D-modules : We have obtained an algorithm for comoputing the (cohomology groups of) the restriction to an affine subvariety, and the integration along an affine space, of a D-module defined on an affine space. This is done by computing a free resolution adapted to some filtration.
2. Algorithms for holonomic functions : It is difficult to treat various functions by computers. Dealing with the differential equations which a function satisfies rather than the function itself, we obtained algorithms for computing the system of differential equtions for the integration and the product of holonomic functions.
3. An algorithm for de Rham cohomology : As an application of the algorithm for computing the integration, we obtained an algorithm to compute the de Rham cohomology groups of the complement of an algebraic set in the affine space.
4. Development and distribution of the software : A system kan, which N.Takayama has been developing from around 1991, is a program for computing D-modules mainly by Groebner basis computation in the ring of differential operators. We have imple- mented the algorihms mentioned above in kan. The programs are freely distributed from http : //www.math.kobe-u.ac.jp/KAN/ along with source codes.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (22 results)

All Other

All Publications (22 results)

  • [Publications] T.Oaku: "Algorithms for the b-function and D-modules" J.Pure Appl.Algebra. 117〜118. 495-518 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Oaku: "Algoritms for b-functions, restrictions・・・" Advances in Appl.Math.19. 61-105 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 大阿久俊則: "計算の視点から見たD加群理論" 数学. 50・2. 203-〓 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 大阿久俊則: "数式処理と数学-ホロノミック・パラダイム" 数理科学. 425. 32-38 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Oaku: "Higher-codimensional boundary value problems" Publ.RIMS, Kyoto Univ.34・3. 383-437 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Oaku: "An algorithms for de Rham cohomology groups" J.Pure Appl.Algebra. (印刷中). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Oaku: "An algorithm of computing b-functions" Duke Math.J.87. 115-132 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Oaku: "Algorithms for the b-function and D-modules associated with a polynomial" J.Pure Appl.Algebra. 117 & 118. 495-518 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Oaku: "Algorithms for b-functions, restrictions, and algebraic local cohomology groups of D-modules" Advances in Applied Math. 19. 61-105 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Oaku, S.Yamazaki: "Higher-codimensional boundary value problems and F-mild microfunctions -local and microlocal uniqueness" Publ.RIMS, Kyoto Univ.34. 383-437 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Oaku, N.Takayama: "An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation" J.Pure Appl.Algebra. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 大阿久 俊則: "加群の積分とその応用" 京都大学数理解析研究所講究録. 1038. 163-169 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 大阿久 俊則: "計算の視点から見たD加群理論" 数学. 50・2. 203-211 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 大阿久 俊則: "数式処理と数学-ホロノミック・パラダイム" 数理科学. 425. 32-38 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Oaku: "Higher-codimensional boundary value problems" Publ.RIMS,Kyoto Univ.34・3. 383-437 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Oaku: "An algorithm for de Rham cohomology groups" J.Pure Appl.Algebra. (印刷中). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Oaku: "An algorithm of computing b-functions" Duke Mathematical Jouranl. 87. 115-132 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Oaku: "Algorithms for the b-function and D-modules associated with a polynomia" Journal of Pure and Applied Algebra. 117. 495-518 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Oaku: "Algorithms for b-functions,restrictions,and algebraic local cohomology" Advances in Applied Mathematics. 19. 61-105 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 大阿久 俊則: "計算の視点から見たD加群理論" 数学. 50.2(印刷中). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Fujii, H.Funahashi: "Extension of the Barut-Girardello coherent states" Journal of Mathematical Physics. 38. 4422-4434 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Fujii, T.Soma: "Totally geodesic bonndaries are dense in the modnli space" Journal of Mathematical Society of Japan. 49. 589-601 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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