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Research of differential or difference equations appearing in many fields and the value distribution of meromorphic functions or holomorphic curves

Research Project

Project/Area Number 19540173
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TOHGE Kazuya  Kanazawa University, 電子情報学系, 准教授 (30260558)

Co-Investigator(Kenkyū-buntansha) 森 正氣  山形大学, 理学部, 教授 (80004456)
下村 俊  慶應義塾大学, 理工学部, 教授 (00154328)
石崎 克也  日本工業大学, 工学部, 准教授 (60202991)
Co-Investigator(Renkei-kenkyūsha) MORI Seiki  山形大学, 理学部, 教授 (80004456)
SHIMOMURA Shun  慶應義塾大学, 理工学部, 教授 (00154328)
ISHIZAKI Katsuya  日本工業大学, 工学部, 教授 (60202991)
Project Period (FY) 2007 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2007: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
KeywordsNevanlinna理論 / 有理型函数 / 正則曲線 / 複素常微分方程式 / 複素差分方程式 / 超離散方程式 / 周期函数 / 一意性問題 / 値分布 / Tropical Nevanlinna理論 / 差分方程式 / 周期関数 / 微分方程式 / Revanlinna理論 / Painleve超越函数 / Painleve方程式 / Schroder方程式 / 各領域の値分布
Research Abstract

複素平面上の有理型函数を係数とする微分、(q-)差分或いは超離散方程式の可積分性を記述する値分布論的な性質として提案されていた位数が有限な有理型函数解の存在条件に対し、超位数が1未満であるというより緩やかな条件が最良であることを示した。またHalburd-Southallによるmax-plusの意味での有理型函数のTropical Nevanlinna理論に於いて第2主要定理の対応物を与えるなどの改良を行った。

Report

(4 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (54 results)

All 2010 2009 2008 2007

All Journal Article (27 results) (of which Peer Reviewed: 22 results) Presentation (25 results) Book (2 results)

  • [Journal Article] On Gundersen's solution to the Fermat-type functional equation of degree 62010

    • Author(s)
      Tohge Kazuya
    • Journal Title

      Complex Variables and Elliptic Equations (in press)

      Pages: 1-18

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] On Gundersen's solution to the Fermat-type functional equation of degree 62010

    • Author(s)
      Kazuya Tohge
    • Journal Title

      Complex Variables and Elliptic Equations (in press)

      Pages: 1-18

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Nevanlinna theory and second main theorem2009

    • Author(s)
      Laine Ilpo, Tohge Kazuya
    • Journal Title

      arXiv. org arXiv:0910. 5300

      Pages: 1-43

    • Related Report
      2009 Final Research Report
  • [Journal Article] Holomorphic curves with shift-invariant hyperplane preimages2009

    • Author(s)
      Halburd Rod, Korhonen Risto, Tohge Kazuya
    • Journal Title

      arXiv. org, arXiv:0903. 3236

      Pages: 1-30

    • Related Report
      2009 Final Research Report
  • [Journal Article] Holomorphic curves with shift-invariant hyperplane preimages2009

    • Author(s)
      Rod Halburd, Risto Korhonen, Kazuya Tohge
    • Journal Title

      arXiv.org arXiv:0903.3236

      Pages: 1-30

    • Related Report
      2009 Annual Research Report
  • [Journal Article] Tropical Nevanlinna theory and second main theorem2009

    • Author(s)
      Ilpo Laine, Kazuya Tohge
    • Journal Title

      arXiv.org arXiv:0910.5300

      Pages: 1-43

    • Related Report
      2009 Annual Research Report
  • [Journal Article] On Gundersen's solution of the Fermat-type functional equation ofdegree $6$2009

    • Author(s)
      Tohge, Kazuya
    • Journal Title

      Complex Variables and Elliptic Equations (掲載確定)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Uniqueness theorems of entire functions with shared-set in an angular domain2008

    • Author(s)
      Lin Weichuan, Mori Seiki
    • Journal Title

      Acta Math. Sinica 24

      Pages: 1925-1934

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Equi-distribution of values for the third and the fifth Painlev\'e transcendents2008

    • Author(s)
      Shimomura Shun
    • Journal Title

      Nagoya Math. J. 192

      Pages: 89-109

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Nonlinear differential equations of second Painlev\'e type with the quasi-Painlev\'e property2008

    • Author(s)
      Shimomura Shun
    • Journal Title

      Tohoku Math. J. 60

      Pages: 581-595

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Remarks on a special fundamental solution base and its product, Complex Analysis and its Applications2008

    • Author(s)
      Tohge Kazuya
    • Journal Title

      Proceedings of the 15th ICFIDCAA Osaka 2007, OCAMI Studies 2

      Pages: 351-356

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Uniqueness theorems of entire functions with shared-set in an angular domain2008

    • Author(s)
      Lin, Weichuan
    • Journal Title

      Acta Math. Sinica 2 4

      Pages: 1925-1934

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers2008

    • Author(s)
      Eisner, Carsten
    • Journal Title

      Ramanujan J. 17

      Pages: 429-446

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Equi-distribution of values for the third and the fifth Painlev\'e transcendents2008

    • Author(s)
      Shimomura, Shun
    • Journal Title

      Nagoya Math. J. 192

      Pages: 89-109

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Nonlinear differential equations of second Painlev\'e type with the quasi-Painlev\'e property2008

    • Author(s)
      Shimomura, Shun
    • Journal Title

      Tohoku Math. J. 60

      Pages: 581-595

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Remarks on a special fundamental solution base and its product2008

    • Author(s)
      Tohge, Kazuya
    • Journal Title

      Complex Analysis and its Applications, Proceedings of the 15th ICFIDCAA Osaka2007, 0CAMI Studies 2

      Pages: 351-356

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Reciprocal sums of Fibonacci numbers2008

    • Author(s)
      Elsner, Carsten
    • Journal Title

      Diophantine Analysis and Related Fields DARF 2007/2008, AIP Conference Proceedings976, American Institute of Physics, Melville, New York

      Pages: 77-89

    • Related Report
      2007 Annual Research Report
  • [Journal Article] On results of H. Ueda and G. Brosch concerning the unicity of meromorphic functions.2007

    • Author(s)
      Qi Han, Mori Seiki, Tohge Kazuya
    • Journal Title

      J. Math. Anal. Appl. 335(2)

      Pages: 915-934

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] On Shared-Value Properties of Painlev\'e Transcendents2007

    • Author(s)
      Lin Weichuan, Tohge Kazuya
    • Journal Title

      Computational Methods and Function Theory 7,2

      Pages: 477-499

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] A class of differential equations of PI-type with the quasi-Painlev\'e property2007

    • Author(s)
      Shimomura Shun
    • Journal Title

      Ann. Mat. Pura Appl. 186

      Pages: 267-280

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Transcendental meromorphic solutions of some algebraic differential equations2007

    • Author(s)
      Ishizaki Katsuya, Toda Nobushige
    • Journal Title

      J. Aust. Math. Soc. 82

      Pages: 1-24

    • NAID

      120006684035

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] On Shared-Value Properties of Painlev\'e Transcendents2007

    • Author(s)
      Lin, Weichuan
    • Journal Title

      Computational Methods and Function Theory 7

      Pages: 447-499

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On results of H.Ueda and G.Brosch concerning the unicity of meromorphic functions2007

    • Author(s)
      Han, Qi
    • Journal Title

      J.Math.Anal.Appl. 335

      Pages: 915-934

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A class of differential equations of PI-type with the quasi-Painlev\'e property2007

    • Author(s)
      Shimomura, Shun
    • Journal Title

      Ann.Mat.Pura Appl. 186

      Pages: 267-280

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic solutions of a degenerate Garnier system of the first Painlev\'e type2007

    • Author(s)
      Shimomura, Shun
    • Journal Title

      RIMS K\^{o}ky\^{u}roku Bessatsu B2

      Pages: 227-237

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Algebraic relations for reciprocal sums of Fibonacci numbers2007

    • Author(s)
      Elsner, Carsten
    • Journal Title

      Acta Arith. 130

      Pages: 37-60

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Transcendental meromorphic solutions of some algebraic differential equations2007

    • Author(s)
      Ishizaki, Katsuya
    • Journal Title

      J.Aust.Math.Soc. 82

      Pages: 1-24

    • NAID

      120006684035

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] 理論とある超離散方程式の解について2010

    • Author(s)
      藤解和也, Laine Ilpo, Tropical Nevanlinna
    • Organizer
      2010年日本数学会年会
    • Place of Presentation
      慶應義塾大学矢上キャンパス(神奈川県)
    • Year and Date
      2010-03-26
    • Related Report
      2009 Final Research Report
  • [Presentation] Tropical Nevanlina 理論とある超離散方程式の解について2010

    • Author(s)
      藤解和也
    • Organizer
      2010年日本数学会年会
    • Place of Presentation
      慶應義塾大学矢上キャンパス (神奈川県)
    • Year and Date
      2010-03-26
    • Related Report
      2009 Annual Research Report
  • [Presentation] Tropical Nevanlinna theory and a second main theorem (joint work with Ilpo Laine)2010

    • Author(s)
      藤解和也
    • Organizer
      複素解析研究集会
    • Place of Presentation
      生涯学習センター遊学館(山形県)
    • Year and Date
      2010-01-09
    • Related Report
      2009 Final Research Report
  • [Presentation] Tropical Nevanlinna theory and a second main theorem (joint work with Ilpo Laine)2010

    • Author(s)
      藤解和也
    • Organizer
      複素解析研究集会
    • Place of Presentation
      生涯学習センター 遊学館 (山形県)
    • Year and Date
      2010-01-09
    • Related Report
      2009 Annual Research Report
  • [Presentation] Tropical Nevanlinna theory and solutions of some ultra-discrete equations (joint work with Ilpo Laine)2009

    • Author(s)
      藤解和也
    • Organizer
      「等角写像論・値分布論」合同研究集会
    • Place of Presentation
      東北大学青葉山キャンパス(宮城県)
    • Year and Date
      2009-12-05
    • Related Report
      2009 Final Research Report
  • [Presentation] Tropical Nevanlinna theory and solutions of some ultra-discrete equations (joint work with Ilpo Laine)2009

    • Author(s)
      藤解和也
    • Organizer
      「等角写像論・値分布論」合同研究集会
    • Place of Presentation
      東北大学青葉山キャンパス (宮城県)
    • Year and Date
      2009-12-05
    • Related Report
      2009 Annual Research Report
  • [Presentation] Holomorphic curves with shift-invariant hyperplane preimages2009

    • Author(s)
      藤解和也
    • Organizer
      2009年度日本数学会秋季総合分科会
    • Place of Presentation
      大阪大学豊中キャンパス(大阪府)
    • Year and Date
      2009-09-26
    • Related Report
      2009 Final Research Report
  • [Presentation] Holomorphic curves with shift-invariant hyperplane preimages2009

    • Author(s)
      藤解和也
    • Organizer
      2009年度日本数学会秋季総合分科会
    • Place of Presentation
      大阪大学豊中キャンパス (大阪府)
    • Year and Date
      2009-09-26
    • Related Report
      2009 Annual Research Report
  • [Presentation] On factorization of meromorphic solutions of some functional equations2008

    • Author(s)
      石崎克也
    • Organizer
      等角写像・値分布論合同研究集会
    • Place of Presentation
      金沢大学サテライト・プラザ(石川県)
    • Year and Date
      2008-11-30
    • Related Report
      2009 Final Research Report
  • [Presentation] On factorization of meromorphic solutions of some functional equations2008

    • Author(s)
      石崎, 克也
    • Organizer
      等角写像・値分布論 合同研究集会
    • Place of Presentation
      金沢大学サテライト・プラザ
    • Year and Date
      2008-11-30
    • Related Report
      2008 Annual Research Report
  • [Presentation] 値分布論入門2008

    • Author(s)
      藤解和也
    • Organizer
      第43回函数論サマーセミナー
    • Place of Presentation
      マホロバマインズ三浦(神奈川県)
    • Year and Date
      2008-08-24
    • Related Report
      2009 Final Research Report
  • [Presentation] 値分布論入門2008

    • Author(s)
      藤解, 和也
    • Organizer
      第43回函数論サマーセミナー
    • Place of Presentation
      マホロバマインズ三浦
    • Year and Date
      2008-08-24
    • Related Report
      2008 Annual Research Report
  • [Presentation] Painleve transcendents in a sectorial domain.2008

    • Author(s)
      Shimomura Shun
    • Organizer
      From Painleve to Okamoto
    • Place of Presentation
      University of Tokyo(東京都)
    • Year and Date
      2008-06-12
    • Related Report
      2009 Final Research Report
  • [Presentation] Painlev\'e transcendents in a sectorial domain2008

    • Author(s)
      Shimomura, Shun
    • Organizer
      From Painlev\'e to Okamoto
    • Place of Presentation
      University of Tokyo
    • Year and Date
      2008-06-12
    • Related Report
      2008 Annual Research Report
  • [Presentation] 角領域における値分布論とその応用2008

    • Author(s)
      下村俊
    • Organizer
      日本数学会年会(函数論分科会特別講演)
    • Place of Presentation
      近畿大学(大阪府)
    • Year and Date
      2008-03-24
    • Related Report
      2009 Final Research Report
  • [Presentation] 角領域における値分布論とその応用2008

    • Author(s)
      下村 俊
    • Organizer
      日本数学会年会(函数論分科会特別講演)
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-24
    • Related Report
      2007 Annual Research Report
  • [Presentation] Holomorphic curves with shift-invariant hyperplane preimages(R. Halburd, R. Korhonenと共同)2008

    • Author(s)
      藤解和也
    • Organizer
      「函数論と値分布論」小澤満先生七年忌追悼研究集会
    • Place of Presentation
      東京工業大学(東京都)
    • Year and Date
      2008-02-09
    • Related Report
      2009 Final Research Report
  • [Presentation] Holomorphic curves with shift-invariant hyperplane preimages(R.Halburd,R.Korhonenと共同)2008

    • Author(s)
      藤解 和也
    • Organizer
      「函数論と値分布論」小澤満先生七年忌追悼研究集会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2008-02-09
    • Related Report
      2007 Annual Research Report
  • [Presentation] Difference operator and value distribution theory2007

    • Author(s)
      石崎克也
    • Organizer
      等角写像値分布論合同研究集会
    • Place of Presentation
      慶應義塾大学(神奈川県)
    • Year and Date
      2007-12-09
    • Related Report
      2009 Final Research Report
  • [Presentation] Difference operator and value distribution theory2007

    • Author(s)
      石崎 克也
    • Organizer
      等角写像値分布論合同研究集会
    • Place of Presentation
      慶鷹義塾大学
    • Year and Date
      2007-12-09
    • Related Report
      2007 Annual Research Report
  • [Presentation] フィボナッチゼータ関数の値について2007

    • Author(s)
      下村 俊
    • Organizer
      短期共同研究「解析的整数論とその周辺」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-10-19
    • Related Report
      2007 Annual Research Report
  • [Presentation] A special fundamental solution base and its product2007

    • Author(s)
      Tohge Kazuya
    • Organizer
      The 15th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications
    • Place of Presentation
      Osaka City University(大阪府)
    • Year and Date
      2007-08-02
    • Related Report
      2009 Final Research Report
  • [Presentation] A special fundamental solution base and its product2007

    • Author(s)
      Tohge, Kazuya
    • Organizer
      The 15th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications
    • Place of Presentation
      Osaka City University
    • Year and Date
      2007-08-02
    • Related Report
      2007 Annual Research Report
  • [Presentation] Topics on meromorphic mappings and defects2007

    • Author(s)
      Mori Seiki
    • Organizer
      Workshop on value distribution theory and its applications
    • Place of Presentation
      Weihai International Center of Shandong University(中国)
    • Related Report
      2009 Final Research Report
  • [Presentation] Topics on meromorphic mappings and defects2007

    • Author(s)
      Mori, Seiki
    • Organizer
      Workshop on value distribution theory and its applications
    • Place of Presentation
      Shandong University
    • Related Report
      2007 Annual Research Report
  • [Book] The Ban k-Laine conjecture-a survey', Some topics on value distribution and di fferentiability in complex and p-adi c analysis, Mathematics Monograph Se ries 112008

    • Author(s)
      Laine Ilpo, Tohge Kazuya
    • Publisher
      Science Press
    • Related Report
      2009 Final Research Report
  • [Book] The Bank-Laine conjecture-a survey', Some topics on value distribution anddifferentiability in complex and p-adic analysis, Mathematics Monograph Series 112008

    • Author(s)
      Laine, Ilpo
    • Publisher
      Science Press
    • Related Report
      2008 Annual Research Report

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Published: 2007-04-01   Modified: 2016-04-21  

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