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

2009 Fiscal Year Final Research Report

• Project/Area Number: 19540173
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kanazawa University
• Principal Investigator: TOHGE Kazuya Kanazawa University, 電子情報学系, 准教授 (30260558)
• 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
• Keywords: Nevanlinna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

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。

Research Products

• [Journal Article] On Gundersen's solution to the Fermat-type functional equation of degree 6 (2010)
  • Author(s): Tohge Kazuya
  • Journal Title: Complex Variables and Elliptic Equations (in press) Pages: 1-18
  • Peer Reviewed
• [Journal Article] Nevanlinna theory and second main theorem (2009)
  • Author(s): Laine Ilpo, Tohge Kazuya
  • Journal Title: arXiv. org arXiv:0910. 5300 Pages: 1-43
• [Journal Article] Holomorphic curves with shift-invariant hyperplane preimages (2009)
  • Author(s): Halburd Rod, Korhonen Risto, Tohge Kazuya
  • Journal Title: arXiv. org, arXiv:0903. 3236 Pages: 1-30
• [Journal Article] Uniqueness theorems of entire functions with shared-set in an angular domain (2008)
  • Author(s): Lin Weichuan, Mori Seiki
  • Journal Title: Acta Math. Sinica 24 Pages: 1925-1934
  • Peer Reviewed
• [Journal Article] Equi-distribution of values for the third and the fifth Painlev\'e transcendents (2008)
  • Author(s): Shimomura Shun
  • Journal Title: Nagoya Math. J. 192 Pages: 89-109
  • Peer Reviewed
• [Journal Article] Nonlinear differential equations of second Painlev\'e type with the quasi-Painlev\'e property (2008)
  • Author(s): Shimomura Shun
  • Journal Title: Tohoku Math. J. 60 Pages: 581-595
  • Peer Reviewed
• [Journal Article] Remarks on a special fundamental solution base and its product, Complex Analysis and its Applications (2008)
  • Author(s): Tohge Kazuya
  • Journal Title: Proceedings of the 15th ICFIDCAA Osaka 2007, OCAMI Studies 2 Pages: 351-356
  • Peer Reviewed
• [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
  • Peer Reviewed
• [Journal Article] On Shared-Value Properties of Painlev\'e Transcendents (2007)
  • Author(s): Lin Weichuan, Tohge Kazuya
  • Journal Title: Computational Methods and Function Theory 7,2 Pages: 477-499
  • Peer Reviewed
• [Journal Article] A class of differential equations of PI-type with the quasi-Painlev\'e property (2007)
  • Author(s): Shimomura Shun
  • Journal Title: Ann. Mat. Pura Appl. 186 Pages: 267-280
  • Peer Reviewed
• [Journal Article] Transcendental meromorphic solutions of some algebraic differential equations (2007)
  • Author(s): Ishizaki Katsuya, Toda Nobushige
  • Journal Title: J. Aust. Math. Soc. 82 Pages: 1-24
  • 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
• [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
• [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
• [Presentation] Holomorphic curves with shift-invariant hyperplane preimages (2009)
  • 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
• [Presentation] On factorization of meromorphic solutions of some functional equations (2008)
  • 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
• [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
• [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
• [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
• [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
• [Presentation] Topics on meromorphic mappings and defects (2007)
  • Author(s): Mori Seiki
  • Organizer: Workshop on value distribution theory and its applications
  • Place of Presentation: Weihai International Center of Shandong University(中国)
  • Year and Date: 20070723-20070729
• [Presentation] Difference operator and value distribution theory (2007)
  • 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
• [Presentation] A special fundamental solution base and its product (2007)
  • 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
• [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 11 (2008)
  • Author(s): Laine Ilpo, Tohge Kazuya
  • Total Pages: 433-453
  • Publisher: Science Press