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Rational singularities, Young diagram, Painleve equation

Research Project

Project/Area Number 11440006
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNagoya University

Principal Investigator

UMEMURA Hiroshi  Nagoya University, Graduate school of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (40022678)

Co-Investigator(Kenkyū-buntansha) NOUMI Masatoshi  Department of Mathematics, Kobe University, Professor, 理学部, 教授 (80164672)
OKAMOTO Kazuo  Graduate school of Mathematical science, University of Tokyo Professor, 大学院・数理科学研究科, 教授 (40011720)
MUKAI Shigeru  Research Institute of Mathematical Science, Kyoto University, Professor, 数理解析研究所, 教授 (80115641)
OKADA Soichi  Nagoya University, Graduate school of Mathematics, Associated Professor, 大学院・多元数理科学研究科, 助教授 (20224016)
浪川 幸彦  名古屋大学, 大学院・多元数理科学研究科, 教授 (20022676)
Project Period (FY) 1999 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥14,500,000 (Direct Cost: ¥14,500,000)
Fiscal Year 2002: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2001: ¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2000: ¥3,900,000 (Direct Cost: ¥3,900,000)
Fiscal Year 1999: ¥4,600,000 (Direct Cost: ¥4,600,000)
KeywordsPainleve equation / Young diagram / Special polynomial / Rational singular puint / Algebrait surface / パンルベ方程式 / 微分ガロア理論 / パーノルベ方程式
Research Abstract

1. Painleve equations and Special polynomial
We disocvered that the Painleve equations generate special polynomials. If we consider the motivation of the discovery of the Painleve equtions, it is surprising that they have combinatorial aspects. We presented conjectures on the special polynomials and proved them.
2. Generalization of the Painleve equations based on the symmetries
Noumi considered that the conjectures should be solved in a natural frame work. To this end, he generalized the Painleve equations from the view point of theory of Lie algebra.
3. Deformation of rational double points and Backhand tranhformations
We proved that the Backhund transformations arise from deformation of rational double points.
4. Infinite dimensional differential Galois theory and Painleve equations
We applied our theory of infinite dimensional to the definition of the Painleve equations.

Report

(5 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (42 results)

All Other

All Publications (42 results)

  • [Publications] 梅村浩, 岡本和夫: "Special polynomials and Hirota bilinear relations of the second and the forth Painleve equation"Nagoya Math. J.. 159. 179-200 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 梅村浩 他: "Painleve equations and deformation of rational surfaces with retional double points"Physics and combinatorics 1999. 320-365 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 岡本和夫 他: "The proof of the Painleve property by Masuo Hukuhara"Funkcial Ekvac. 44. 201-217 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 向井茂: "Geometric realization of T-shaped root systems and counter examples to Hilbert 14th problem"Kyoto Univ, RIMS, Preprint. (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 野海正俊 他: "Symmetries in the fourth Painleve equation and Okamoto polynomiouls"Nagoya Math. J.. 153. 53-86 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 野海正俊 他: "A new Lax pair for the sixth Painleve equation associated with so(8)"Microlocal Analysis and complex Fourier Analysis. 238-252 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 梅村 浩: "楕円関数論"東大出版会. 362 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroshi Umemura, Kazuo Okamoto, et al: "Special polynomials and Hirota bilinear relations of the second and the fourth Puinleve equations"Nagoya Math.J.. 159. 179-200 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroshi Umemura et al.: "Painleve equations and defovmation of rational smfaces milh rational double points"Physics and Combinatorics 1999 World Scientifiu. 320-365 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Okamoto et al.: "The puof of the Painleve propeit. Masuo Hukuhara"Funkcial.Ekuac. 44. 201-217 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Shigeru Mukai: "Geometue realizati on of T-shaped root systems and counter examples to Hilbert 14th probleme"Kyoto University, RIMS Preprint.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masatoshi Noumi et al.: "Symmetries in the fourth Painleve equations 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
      2002 Final Research Report Summary
  • [Publications] Masatoshi Noumi et al.: "Anew Lax pair for thr sixth Painleve equation associated with solgl"Microlocal Analysis and Fourun Analysis, World Sci.. 238-252 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Umemura: "Thcory of elliptie functions, Unirity of Tokyo Press"362 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 梅村 浩: "On the definitions of the Painleve equations"京都大学数理科学研究所講究録. 1296. (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 向井 茂: "Geometric realization of T-shaped root systems and counter examples to Hilbert 14th problem"RIMS, Kyoto Univ. Preprint. 1372. (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 野海 正俊: "Affine weyl group approach to Painleve equations"Proc. ICM, 2002 Beijing. 497-509 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 野海 正俊 他: "Backlund transformations and the manifolds of Painleve systems"Funkcial. Ekvac.. 45. 237-258 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 野海 正俊 他: "Discrete dynamical systems with W(A^<(1)>_<m-1>×A^<(1)>_<n-1>)symmetry"Lett. Math. Phys.. 60. 211-219 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 野海 正俊 他: "A new Lax pain for the sixth Painleve equation asociated with <so>^^^^(8)"Microlocal Analysis and Fourier Analysis. 238-252 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 野海正俊, 山田泰彦: "Painleve方程式の対称性について"数学. 53. 62-75 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Noumi et al.: "Determinant formula for the Toda and discrete Toda equations"Funkcial. Ekvac.. 44. 291-307 (2001)

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

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Taneda: "Polynomials associated with an aljebraic solution of the sixth Painleve equation"Jap. J. Math.. 27. (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Kirillov, M.Taneda: "Generalized Umemura polynomials"Rocky Mountain J. of Math.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] A. Kirillov, M. Taneda: "Generalized Umemura polynomials and Hirota-Miwa equations"MSJ Memoirs. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Umemura et al.: "Special polynomials nelated with the second and the Sourth Pounleve equationw"Nagoya Math.J.. 159. 179-200 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H,Umemuca et al.: "Painleve equations and deformations of rational susfaces with rational double points"Proc.Symp.on Physics and Combinatrics 1999. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Noumi et al.: "Raising operators of row type for Macclonald polynomials"Composition Math.. 120. 119-136 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Noumi et al.: "Determinant formulas forth Toda and discrete Toda equations"Funkcial,Ekvac.. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Noumi et al.: "Birational Weyl group action arising from a noilpotent Poisson algema"Proc,Symp.on Physics and Comtinatoucs 1999. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] M,Noumi et al.: "Tableau representation for Macclonald's ninth vauation of Schur functions"Proe.Symo on Physics and Conbinatorics 2000. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 梅村浩: "楕円関数論"東大出版. 362 (2000)

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

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Umemura: "Painleve方程式の100年"数学. 51. 395-420 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Umemura: "On the transformation group of the second Painleve equation"Nagoya Math. J.. 157. 1-32 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Umemura et al.: "Special polynomials and the Hirota bilinear relations of the 2nd and the 4th Painleve equations"Nagoya Math. J. (発表予定).

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

    • Related Report
      1999 Annual Research Report
  • [Publications] A.N.Kirillov and M.Noumi: "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] Y.Kajiwara and Noumi: "Raising operators of row types for Macdonald polynomials"Compositio Mathematica. 120. 119-136 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 梅村 浩: "楕円関数論"東大出版 (出版予定). 359 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 向井 茂: "モジュライ理論2"岩波書店 (出版予定). (2000)

    • Related Report
      1999 Annual Research Report

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

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