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Comprehensive Studies on Hypergeometric and Painleve Systems

Research Project

Project/Area Number 11304007
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKobe University

Principal Investigator

NOUMI Masatoshi  Kobe University, Graduate School of Science and Technology, Professor, 大学院・自然科学研究科, 教授 (80164672)

Co-Investigator(Kenkyū-buntansha) SAITO Masahiko  Faculty of Science, Professor, 理学部, 教授 (80183044)
SASAKI Takeshi  Faculty of Science, Professor, 理学部, 教授 (00022682)
TAKANO Kyoichi  Faculty of Science, Professor, 理学部, 教授 (10011678)
MASUDA Tetsu  Graduate School of Science and Technology, Assistant, 大学院・自然科学研究科, 助手 (00335457)
YAMADA Yasuhiko  Faculty of Science, Professor, 理学部, 教授 (00202383)
高山 信毅  神戸大学, 理学部, 教授 (30188099)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥20,990,000 (Direct Cost: ¥19,700,000、Indirect Cost: ¥1,290,000)
Fiscal Year 2001: ¥5,590,000 (Direct Cost: ¥4,300,000、Indirect Cost: ¥1,290,000)
Fiscal Year 2000: ¥7,400,000 (Direct Cost: ¥7,400,000)
Fiscal Year 1999: ¥8,000,000 (Direct Cost: ¥8,000,000)
KeywordsPainleve equation / affine Weyl group symmetry / hypergeometric equation / configuration space / integrable system / ベックルント変換 / 離散可積分系 / 初期値空間 / モジュライ空間 / ミラー対称性 / ハミルトン構造
Research Abstract

Main achievements of this research are summarized as follows.
1. Affine Weyl group symmetry of Painleve systems: Generalizing the structure of Backlund transformations for Painleve equations, Noumi and Yamada proposed a universal framework of Weyl group symmetry as groups of birational transformations. This class of groups of birational transformations is defined in terms of root systems, and has its origin in Kac-Moody groups. Also, they found a method for constructing a class of nonlinear (partial) differential equations of Painleve type by similarity reduction from the infinite integrable systems of Drinfeld-Sokolov. This work brought out a new unified point of view on the Lie theoretic background of the symmetry of Painleve equations.
2. Geometric aspects of hypergeometric and Painleve systems: In the joint work with Yoshida, Sasaki constructed the uniformizing differential equation for the moduli space of cubic surfaces. This system of linear differential equations, with Weyl group symmetry of type E_6, arises as a subsystem of the hypergeometric system of type E(3, 6) associated with hyperplane configurations. On the basis of Sakai's geometric approach to Painleve equations, Saito and others found a method of constructing the Painleve equations and their defining manifolds from the algebraic- geometric theory of deformations of rational surfaces.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (86 results)

All Other

All Publications (86 results)

  • [Publications] M.Noumi, Y.Yamada: "Symmetries in the fourth Painleve equation and Okamoto polynomials"Nagoya Math.J.. 153. 53-86 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Yamada: "Determinant formulas for the τ-functions of the Painleve equations of type A"Nagoya Math.J.. 156. 123-134 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Matano et al.: "On some Hamiltonian structures of Painleve systems. II"J.Math.Soc.Japan. 51. 843-866 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Sasaki, M.Yoshida: "The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces"Proc.Japan Acad.Ser.A Math.Sci.. 75. 129-133 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 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. 139. 201-233 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Saito et al.: "Hypergeometric polynomials and integer programming"Compositio Math.. 115. 185-204 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Noumi, Y.Yamada: "Affine Weyl group symmetries in Painleve type equations"Toward the exact WKB analysis of differential equations, linear or non-linear, Kyoto University Press. 245-259 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Yamada: "Special polynomials and generalized Painleve equations"Adv.Stud.Pure Math.. 28. 391-400 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Takano: "Defining manifolds for Painleve equations"Toward the exact WKB analysis of differential equations, linear or non-linear, Kyoto University Press. 261-269 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Sasaki, M.Yoshida: "A system of differential equations in 4 variables of rank 5 invariant under the Weyl group of type E_6"Kobe J. Math.. 17. 29-57 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Majima et al.: "Quadratic relations for confluent hypergeometric functions"Tohoku Math.J.. 52. 489-513 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Oaku et al.: "A localization algorithm for D-modules"J.Symbolic Comput.. 29. 721-728 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Noumi, Y.Yamada: "Birational Weyl group action arising from a noilpotent Poisson algebra"Physics and Combinatorics 1999, World Scientific. 289-319 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Kajiwara et al.: "Determinant formulas for the Toda and descrete Toda equations"Funkcial.Ekvac.. 44. 291-307 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Kajiwara et al.: "A study on the fourth q-Painleve equation"J.Phys. A : Math.Gen.. 34. 8563-8581 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Yamada: "A birational representation of Weyl group, combinatorial R-matrix and discrete Toda equation"Physics and Combinatorics, 2000, World Scientific. 305-319 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Takano: "Confluence processes in defining manifolds for Painleve systems"Tohoku Math.J.. 53. 319-335 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Sasaki, M.Yoshida: "A geometric study of the hypergeometric function with imaginary exponents"Experiment.Math.. 10. 321-330 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Sasaki, M.Yoshida: "The uniformizing differential equation of the complex hyperbolic structure on the moduli space of a marked cubic surface.II"J.Phys.A. 34. 2319-2328 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.-H.Saito, H.Umemura: "Painleve equations and deformations of rational surfaces with rational double points"Physics and Combinatorics 1999, World Scientific. 320-365 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Oaku, N.Takayama: "Algorithms for D-modules-restriction, tensor product, localization, and local cohomology groups"J.Pure Appl.Algebra. 156. 267-308 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 野海 正俊: "パンルヴェ方程式-対称性からの入門"朝倉書店. 204 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Saito, B.Sturmfels, N.Takayama: "Grobner deformations of hypergeometric differential equations"Springer-Verlag, Berlin. viii+254 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Noumi, and Y. Yamada: "Symmetries in the fourth Painleve equation and Okamoto polynomials"Nagoya Math. J.. 153. 53-86 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y. Yamada: "Determinant formulas for the γ-functions of the Painleve equations of type A"Nagoya Math. J.. 156. 123-134 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Matano, A. Matumiya, and K. Takano: "On some Hamiltonian structures of Painleve systems. II"J. Math. Soc. Japan. 51, no. 4. 843-866 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Sasaki, and M. Yoshida: "The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces"Proc. Japan Acad. Ser. A Math. Sci.. 75, no. 7. 129-133 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Oaku, and N. Takayama: "An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation"J. pure Appl. Algebra. 139, no. 1-3. 201-233 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Saito, B. Sturmfels, and N. Takayama: "Hypergeometric polynomials and integer programming"Compositio Math.. 115, no. 2. 185-204 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Sasaki, and M. Yoshida: "A system of differential equations in 4 variables of rank 5 invariant under the Weyl group of type E_6"Kobe J. Math.. 17, no. 1. 29-57 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Majima, K. Matsumoto, and N. Takayama: "Quadriatic relations for confluent hypergeometric functions"Tohoku Math. J. (2). 52, no. 4. 489-513 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Oaku, N. Takayama, and U. Walther: "A localization algorithm for D-modules"J. Symbolic Comput.. 29, no. 4-5. 721-728 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, and Y. Yamada: "Determinant formulas for the Toda and discrete Toda equations"Funkcial. Ekvac.. 44. 291-307 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Kajiwara, M. Noumi, and Y. Yamada: "A study on the fourth q-Painleve equation"J. Phys. A : Math. Gen.. 34. 8563-8581 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Takano: "Confluence processes in defining manifolds for Painleve systems"Tohoku Math. J. (2). 53, no. 2. 319-335 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Sasaki, and M. Yoshida: "A geometric study of the hypergeometric function with imaginary exponents"Experiment. Math.. 10, no. 3. 321-330 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Sasaki, and M. Yoshida: "The uniformizing differential equation of the complex hyperbolic structure on the moduli space of a marked cubic surface. II"J. Phys. A. 34, no. 11. 2319-2328 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Oaku, and N. Takayama: "Algorithms for D-modules-restriction, tensor, product, localization, and local cohomology groups"J. Pure Appl. Algebra. 156, no. 2-3. 267-308 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Noumi, and Y. Yamada: "Affine Weyl group symmetries in Painleve type equations in: Toward the exact WKB analysis of differential equations, linear or non-linear"Kyoto University Press. 245-259 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y. Yamada: "Special polynomials and generalized Painleve equations in: Combinatorial methods in representation theory, Adv. Stud. Pure Math. 28"Kinokuniya. 391-400 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Takano: "Defining manifolds for Painleve equations in: Toward the exact WKB analysis of differential equations, linear or non-linear"Kyoto University Press. 261-269 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Noumi, and Y. Yamada: "Birational Weyl group action arising from a nilpotent Poisson algebra in: Physics and Combinatorics 1999"World Scientific. 287-319 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y. Yamada: "A birational representation of Weyl group, combinatorial R-matrix and discrete Toda equation in: Physics and Combinatorics 2000"World Scientific. 305-319 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.-H. Saito, and H. Umemura: "Painleve equations and deformations of rational surfaces with rational double points in: Physics and Combinatorics 1999"World Scientific. 320-365 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Noumi: "Painleve equations - introduction through symmetry (in Japanese)"Asakura Shoten. 204 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Saito, and B. Sturmfels, and N. Takayama: "Grobner deformations of hypergeometric differential equations in: Algorithms and Computation in Mathematics 6"Springer-Verlag. 254 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Noumi, Y.Yamada: "Birational Weyl group action arising from a noilpotent Poisson algebra"Physics and Combinatorics 1999, World Scientific. 287-319 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] J.Nakagawa et al.: "Tableau representation for Macdonald's ninth variation of Schur functions"Physics and Combinatorics 2000, World Scientific. 180-195 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Kajiwara et al.: "Determinant formulas for the Toda and descrete Toda equations"Funkcial. Ekvac.. 44. 291-307 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Kajiwara et al.: "A study on the fourth q-Painleve equation"J. Phys. A : Math. Gen.. 34. 8563-8581 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Noumi et al.: "Backlund transformations and the manifolds of Painleve systems"Funkcial. Ekvac.. (to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.-H.Saito et al.: "Deformation of Okamoto-Painleve Pairs and Painleve equations"J. Algebraic Geom.. 11. 311-362 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Takano: "Confluences of Defining Manifolds of Painleve Systems"Tohoku Math. J.. 53. 319-335 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Sasaki: "Projective surfaces defined by Appell's hypergeometric systems E4 and E2"Kyushu J. Math.. 55. 1-21 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Sasaki, M.Yoshida: "A geometric study of the hypergeometric function with imaginary exponents"Experimental Math.. 10. 321-330 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Yamada: "A birational representation of Weyl group, combinatorial R-matrix and discrete Toda equation"Physics and Combinatonics 2000, World Scientific. 305-319 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Masuda: "On the rational solutions of q-Painleve V equation"Nagoya Math. J.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Sekiguchi: "Combinatorial formula of the dimension of global solutions to a generalized hypergeometric systems Μ^^〜_<3,2>(ν)"Japan. J. Math. (N.S.). 27. 311-326 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Sekiguchi: "The Penrose transform for Sp(n, R) and singular unitary representations"J. Math. Soc. Japan. 54. 215-253 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Noumi,Masatoshi: "Affine Weyl group symmetries in Painleve type equations""Toward the exact WKB analysis of differential equations, linear or non-linear", Kyoto University Press. 245-259 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kajihara,Yasuhi: "Raising operators of row type for Macdonald polynomials"Compositio Math.. 120. 119-136 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kajiwara,Kenji: "Determinant formulas for the Toda and descrete Toda equations"Funkcial.Ekvac.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Noumi,Masatoshi: "Birational Weyl group action arising from a noilpotent Poisson algebra"Proceedings of the International Workshop on Physics and Combinatorics 1999, Nagoya. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Nakagawa,Jun: "Tableau representation for Macdonald's ninth variation of Schur functions"Proceedings of the International Workshop on Physics and Combinatorics 2000, Nagoya. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Sasaki,Takeshi: "A system of differential equations in 4 variables of rank 5 invariant under the Wey group of type E_6"Kobe J.Math.. 17. 29-57 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Sasaki,Takeshi: "Projective surfaces defined by Appell's hypergeometric systems E_4 and E_2"Kyushu J.Math.. 55. 1-21 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Sasaki,Takeshi: "The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces, II"J.Phys.A. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Saito,Masa-Hiko: "Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and mirror symmetry""The arithmetic and geometry of algebraic cycles" (Banff,AB,1998), Kluwer Acad, Publ.. 385-425 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Majima,Hideyuki: "Quadratic relations for confluent hypergeometric functions"Tohoku Math.J.. 52. 489-513 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Oaku,Toshinori: "A localization algorithm for D-modules"J.Symbolic Comput.29. 29. 721-728 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Takano,Kyoichi: "Defining manifolds for Painleve equations""Toward the exact WKB analysis of differential equations, linear or non-linear" (Kyoto,1998), Kyoto University Press. 261-269 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Fukae,Mitsuaki: "Mordell-Weil lattice via string junctions"Nuclear Phys.B. 572. 71-94 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Yamada,Yasuhiko: "A birational representation of Weyl group, combinatorial R-matrix and discrete Toda equation"Proceedings of the International Workshop on Physics and Combinatorics 2000, Nagoya. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] 野海正俊: "パンルヴェ方程式-対称性からの入門"朝倉書店. 204 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 高野恭一: "パンルヴェ方程式の眺望"神戸大学理学部数学教室. 192 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Saito,Mutsumi: "Grobner deformations of hypergeometric differential equations"Springer-Verlag, Berlin. 262 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Noumi,Masatoshi: "Symmetries in the fourth Painlev equation and Okamoto polynomials"Nagoya Math.J.. 153. 53-86 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Kirillov,Anatol: "q-Difference raising operators for Macdonald polynomials and the integrality of transition coefficients"CRM Proceedings and Lecture Notes. 22. 227-243 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Kajihara,Yasushi: "Raising operators of row type for Macdonald polynomials"Compositio Math.. 120. 119-136 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Sasaki,Takeshi: "The uniformizing differential equation of the complex hypergolic structure on the moduli space of marked cubic surfaces"Proc.Japan Acad.. 75. 129-133 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Nomizu,Katsuhisa: "Globally defined linear connections on the real line and the circle"Tohoku Math.J.. 51. 205-212 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Saito,Masa-Hiko: "Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface"Adv.Theor.Math.Phys.. 3. 177-208 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Saito,Masa-Hiko: "On Mordell-Weil lattices of nonhyperelliptic type"Dokl.Akad.Nauk. 364. 596-598 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Takayama,Nobuki: "An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation"J.Pure Appl.Algebra. 139. 201-233 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Saito,Mutsumi: "Hypergeometric polynomials and Integer Programming"Compositio Math.. 115. 185-204 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Yamada,Yasuhiko: "Determinant formulas for the γ-functions of the Painleve equations of type A"Nagoya Math.J.. 156. 123-134 (1999)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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