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Computational Analysis of Hypergeometric Differential Equations

Research Project

Project/Area Number 15340045
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKOBE UNIVERSITY

Principal Investigator

TAKAYAMA Nobuki  Kobe University, Faculty of Science, Professor, 理学部, 教授 (30188099)

Co-Investigator(Kenkyū-buntansha) NORO Masayuki  Kobe University, Faculty of Science, Professor, 理学部, 教授 (50332755)
FUKUYAMA Katsushi  Kobe University, Faculty of Science, Professor, 理学部, 教授 (60218956)
MASUDA Tetsu  Kyoto University, Graduate School of Science, COE Researche, 理学研究科, COE研究員 (00335457)
OAKU Toshinori  Tokyo Women's Christian University, Faculty of Literature and Science, Professor, 文理学部, 教授 (60152039)
SAITO Mutsumi  Hokaido University, Graduate School of Science, Associate Profsor, 理学研究科, 助教授 (70215565)
松本 圭司  北海道大学, 理学研究科, 助教授 (30229546)
Project Period (FY) 2003 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥10,200,000 (Direct Cost: ¥10,200,000)
Fiscal Year 2006: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2005: ¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2004: ¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2003: ¥2,300,000 (Direct Cost: ¥2,300,000)
KeywordsA-hypergeometric function / A-hypergeometric equation / D-modules / Computational D-modules / Grobner basis / Mathematical formula / Special function / random numbers and integration / calA-超幾何関数 / calA-超幾何方程式 / 超幾何関数の積分表示
Research Abstract

We have obtained the following results
1. We constructed vol(A)-linearly independent convergent series solutions for A-hypergeometric differential-difference equations.
2. We prove that any local Grober fan is a polyhedral fan. As an application of this fact, we give an algorithm of computing local BS polynomials, that of computing local tropical varieties, and discuss a relation of slopes and local Grobner fan.
3. We gave a tangent cone algorithm to study D-modules locally.
We have more results. As to these, refer to Japanese version of this research report and papers.

Report

(5 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (32 results)

All 2007 2006 2005 2004 2003 Other

All Journal Article (24 results) Book (2 results) Publications (6 results)

  • [Journal Article] Local Grobner Fan2007

    • Author(s)
      R.Bahloul, N.Takayama
    • Journal Title

      C.R. Acad.Sci. Paris, Ser. I. (to appear)

    • NAID

      120001506675

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Primitive ideals of the ring of differential operators on an affine toric variety2007

    • Author(s)
      M.Saito
    • Journal Title

      Tohoku Mathematical Journal (to appear)

    • NAID

      110006196742

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Primitive ideals of the ring of differential operators on an affine toric variety2007

    • Author(s)
      M.Snito
    • Journal Title

      Tohoku Mathematical Journal (to appear)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Algebraic relations among some theta functions on the bounded symmetric domain of type I_<γ,γ>.2006

    • Author(s)
      K.Matsumoto
    • Journal Title

      Kyushu J. Math. 60

      Pages: 63-77

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] q-Painleve VI equation arising from q-UC hierarchy2006

    • Author(s)
      T.Tsuda, T.Masuda
    • Journal Title

      Comm. Math. Phys. 262

      Pages: 595-609

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Gap series and functions of bounded variation2006

    • Author(s)
      K.Fukuyama
    • Journal Title

      Acta Math. Hungar. 110

      Pages: 175-191

    • NAID

      120005980769

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] An Efficient Implementation for Computing Groebner Bases over Algebraic Number Fields2006

    • Author(s)
      M.Noro
    • Journal Title

      Proceedings of ICMS 2006, LNCS 4151

      Pages: 99-109

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Algebraic relations among some theta functions on the bounded symmetric domain of type2006

    • Author(s)
      K.Matsumoto
    • Journal Title

      Kyushu J. Math. 60

      Pages: 63-77

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Gap series and functions of bounded variation,2006

    • Author(s)
      K.Fukuyama
    • Journal Title

      Acta Math. Hungar. 110

      Pages: 175-191

    • NAID

      120005980769

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Andres Iglesias edited, Proceedings of ICMS 2006, Lecture Notes in Computer Science (LNCS)2006

    • Author(s)
      Nobuki Takayama
    • Journal Title

      LNCS 4151

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Algebraic relations among some theta functions on the bounded symmetric domain of type I_<r,r>2006

    • Author(s)
      K.Matsumoto
    • Journal Title

      Kyushu J. Math. 60

      Pages: 63-77

    • NAID

      130000063156

    • Related Report
      2006 Annual Research Report
  • [Journal Article] q-Painleve VI equation arising from q- UC hierarchy2006

    • Author(s)
      T.Tsuda, T.Masuda
    • Journal Title

      Comm. Math. Phys. 262

      Pages: 595-609

    • Related Report
      2006 Annual Research Report
  • [Journal Article] An Efficient Implementation for Compating Groebner Bases over Algebraic Number Fields2006

    • Author(s)
      M.Noro
    • Journal Title

      Proceedings of ICMS 2006, LNCS 4151

      Pages: 99-109

    • Related Report
      2006 Annual Research Report
  • [Journal Article] OpenXMプロジェクトの過去,現在,未来2005

    • Author(s)
      高山信毅
    • Journal Title

      数式処理 12

      Pages: 51-56

    • NAID

      10019341853

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Tangent cone algorithm for homogenenized differential operators2005

    • Author(s)
      M.Granger, T.Oaku, N.Takayama.
    • Journal Title

      Journal of Symbolic Computation 39

      Pages: 417-431

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Special polynomials associated with the Noumi-Yamada system of type A_5^<(1)>2005

    • Author(s)
      T.Masuda
    • Journal Title

      Funkcialaj Ekvacioj 48

      Pages: 231-246

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Existence of the exact CNOT on a quantum computer with the exchange interaction2005

    • Author(s)
      Y.Kawano, K.Kimura, M.Noro
    • Journal Title

      Quantum Information Processing 4(2)

      Pages: 65-85

    • Related Report
      2005 Annual Research Report
  • [Journal Article] ディジタル数学公式集の現状と未来2004

    • Author(s)
      高山信毅
    • Journal Title

      情報処理

      Pages: 740-745

    • NAID

      110002768272

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Minimal filtered free resolutions for analytic D-modules2004

    • Author(s)
      M.Granger, T.Oaku
    • Journal Title

      Journal of Pure and Applied Algebra 191

      Pages: 157-180

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Finite generation of rings of differential operators of semigroup algebras2004

    • Author(s)
      M.Saito, W.N.Traves
    • Journal Title

      Journal of Algebra 278

      Pages: 76-103

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Theta constants associated to coverings of P^1 branching at eight points2004

    • Author(s)
      K.Matsumoto, H.Terasoma
    • Journal Title

      Compositio Mathematicae 140

      Pages: 1277-1301

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Implementation of prime decomposition of polynomial ideals over small finite fields2004

    • Author(s)
      M.Noro, K.Yokoyama
    • Journal Title

      Journal of Symbolic Computation 38

      Pages: 1227-1246

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Classical transcendental solutions of the Painleve equations and their degeneration2004

    • Author(s)
      T.Masuda
    • Journal Title

      Tohoku Mathematical Journal 56

      Pages: 467-490

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Takayama, Algebraic algorithms for D-modules and numerical analysis2003

    • Author(s)
      T.Oaku, Y.Shiraki, N
    • Journal Title

      Computer mathematics, Lecture Notes Ser. Comput.(World Sci. Publishing) 10

      Pages: 23-39

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Proceedings of ICMS 2006, Lecture Notes in Computer Science (LNCS) 4151.2006

    • Author(s)
      Nobuki Takayama, Andres Iglesias編集
    • Total Pages
      452
    • Publisher
      Springer
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Proceedings of ICMS 2006, Lecture Notes in Computer Science (LNCS) 4151.2006

    • Author(s)
      Nobuki Takayama, Andres Iglesias 編集
    • Total Pages
      452
    • Publisher
      Springer
    • Related Report
      2006 Annual Research Report
  • [Publications] N.Takayama: "Generating Kummer Type Formulas for Hypergeometric Functions"Algebra, Geometry, and Software Systems. 131-145 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] F.Castro-Jimenez, N.Takayama: "Singularities of the Hypergeometric System associated with a Monomial Curve"Transaction of the American Mathematica Society. 355. 3761-3775 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Oaku, Y.Shiraki, N.Takayama: "Algorithms for D-modules and Numerical Analysis"Proceedings of ASCM 2003. 23-39 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Noro: "A Computer Algebra System : Risa/Asir"Algebra, Geometry, and Software Systems. 147-162 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Masuda: "On a class of algebraic solutions to the Painleve VI equation, its determinant formula"Funkcialaj Ekvacioj. 46. 121-171 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Matsumoto, H.Terasoma: "Theta constants associated to cubic threefolds"Journal of Algebraic Geometry. 12. 741-775 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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