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2005 Fiscal Year Final Research Report Summary

Realization of decompositions of algebraic structures on real computer

Research Project

Project/Area Number 15340011
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionRikkyo University (2005)
Kyushu University (2003-2004)

Principal Investigator

YOKOYAMA Kazuhiro  Rikkyo University, Department of Mathematics, Professor, 理学部, 教授 (30333454)

Co-Investigator(Kenkyū-buntansha) NORO Masayuki  Kobe University, Department of Mathematics, Professor, 理学部, 教授 (50332755)
Project Period (FY) 2003 – 2005
KeywordsComputer Algebra / Groebner Base / Computer Assisted Proof / Computer Algebra System / Splitting Field / Quantifier Elimination / Numeric-Symbolic Computation / Vertex Operator Algebra
Research Abstract

The goal of the project consists of two aims : The first aim is to examine how high/deep mathematical operations on algebraic structures can be executed on real computer by concentrating on decompositions of algebraic structures. Using symbolic and algebraic computation, we try to realize the mathematical operations related to "decomposition". The second one is to utilize realized operations for studies on mathematics as computational tools. The realized mathematical operations on computer shall support mathematicians to investigate unsolved problems and it can produce certain computer-assisted-proofs. Extending the ability of such computations, we apply those to real engineering problems. For those aims, we selected a number of themes, for which we developed effective/efficient algorithms, implemented those, and examined their ability on computational experiments. We have obtained satisfactory results and found promising approaches for realization of more higher/deeper mathematical op … More erations. We list themes and give details for each :
(1)Commutative Algebra :
For prime decomposition of polynomial ideals over finite fields, we obtained an efficient algorithm and its practical implementation. For polynomial ideals with parametric exponents, we defined certain stability of those ideals based on "forms of Groebner bases", and gave a complete algorithm for deciding such stability for simpler cases. We applied "numeric-symbolic computation" to the CAD algorithm for quantifier elimination.
(2)Commutative Algebra with High Symmetry :
We obtained a practical method for computing the splitting field of a polynomial with rational coefficients by using p-adic approximations of its roots and information of its Galois group.
(3)Non-Commutative Algebra :
We obtained a computer-assisted-proof in classification of irreducible modules of the vertex operator algebra derived from a lattice.
(4)Supports for Mathematics and Engineering :
We also obtained a computer-assisted-proof in solving an unsolved conjecture related algebraic curves, and we also applied Groebner bases technique successfully to problems in control theory. Less

  • Research Products

    (20 results)

All 2006 2005 2004 2003 Other

All Journal Article (19 results) Book (1 results)

  • [Journal Article] On polynomial curves in the affine plane2006

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

      Osaka Journal of Mathematics (to appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A modular method for computing the splitting field of a polynomial2006

    • Author(s)
      G.Renault, K.Yokoyama
    • Journal Title

      Proceedings of the 7th Algorithmic Number Theory Symposium (ANTS VII) (to appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Modular dynamic evaluation2006

    • Author(s)
      M.Noro
    • Journal Title

      Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation (ISSAC 2006) (to appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Cylindrical algebraic decomposition via numerical computation with validated symbolic reconstruction2005

    • Author(s)
      H.Anai, K.Yokoyama
    • Journal Title

      Proceedings of Algorithmic Algebra and Logic (A3L2005)

      Pages: 25-30

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Sum of roots with positive real parts2005

    • Author(s)
      H.Anai, S.Hara, K.Yokoyama
    • Journal Title

      Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation (ISSAC 2005)

      Pages: 21-28

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Existence of the exact CNOT on a quantum computer with the exchange interaction2005

    • Author(s)
      Y.Kawano et al.
    • Journal Title

      Quantum Information Processing 4

      Pages: 65-85

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 離散戸田方程式を用いた大規模行列の連立一次方程式,行列式,固有多項式の計算法2005

    • Author(s)
      木村欣司, 野呂正行, 辻本諭, 中村佳正
    • Journal Title

      日本応用数理学会論文誌 15

      Pages: 307-322

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Dynamic Evaluation の実装について2005

    • Author(s)
      野呂正行
    • Journal Title

      数式処理 11

      Pages: 21-28

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 関孝和の問題を解く2005

    • Author(s)
      木村欣司, 平野照比古, 横山和弘
    • Journal Title

      数式処理

      Pages: 35-42

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Existence of the exact CNOT on a quantum computer with the exchange interaction2005

    • Author(s)
      Y.Kawano, K.Kimura, H.Sekigawa, M.Noro, K.Shirayanagi, M.Kitagawa, M.Ozawa
    • Journal Title

      Quantum Information Processing 4

      Pages: 65-85

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On systems of algebraic equations with parametric exponents2004

    • Author(s)
      K.Yokoyama
    • Journal Title

      Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (ISSAC 2004)

      Pages: 312-319

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Z_3 symmetry and W_3 algebra in lattice vertex operator algebras2004

    • Author(s)
      C.Dong et al.
    • Journal Title

      Pacific Journal of Mathematics 215

      Pages: 245-296

    • Description
      「研究成果報告書概要(和文)」より
  • [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

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On systems of algebraic equations with parametric exponents2004

    • Author(s)
      K.Yokoyama
    • Journal Title

      Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation(ISSAC 2004)

      Pages: 312-319

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Z_3 symmetry and W_3 algebra in lattice vertex operator algebras2004

    • Author(s)
      C.Dong, C.H.Lam, K.Tanabe, H.Yamada, K.Yokoyama
    • Journal Title

      Pacific Journal of Mathematics 215

      Pages: 245-296

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Implementation of prime decomposition of polynomial ideals over small finite field2004

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

      Journal of Symbolic Computation 38

      Pages: 1227-1246

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On polynomial curves in the affine plane

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

      Osaka Journal of Mathematics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A modular method for computing the splitting field of a polynominal

    • Author(s)
      G.Renault, K.Yokoyama
    • Journal Title

      Proceedings of the 7th Algorithmic Number Theory Symposium (ANTS VII) (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Modular dynamic evaluation

    • Author(s)
      M.Noro
    • Journal Title

      Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation (ISSAC 2006) (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] グレブナー基底の計算 基礎篇2003

    • Author(s)
      野呂正行, 横山和弘
    • Total Pages
      288
    • Publisher
      東京大学出版会
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2007-12-13  

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