The adaptation of complex function theory to ultradiscrete function theory modeled on value distribution theories and its application to various fields

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05194
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kanazawa University
• Principal Investigator: Tohge Kazuya 金沢大学, 電子情報通信学系, 教授 (30260558)
• Research Collaborator: Korhonen Risto University of Eastern Finland, Department of Physics and Mathematics, Professor ; Laine Ilpo University of Eastern Finland, Department of Mathematics, Professor emeritus ; Heittokangas Janne University of Eastern Finland, Department of Physics and Mathematics, Associate Professor ; Ishizaki Katsuya 放送大学, 教授 ; Gundersen Gary G. University of New Orleans, Department of Mathematics, Professor ; Steinmetz Norbert Technische Universität Dortmund, Institut für Mathematik, Professor emeritus ; Zheng Jian-Hua Tsinghua University, Department of Mathematical Sciences, Professor ; Lin Wei-Chuan Fujian Normal University, Department of Mathematics, Professor ; Wen Zhi-Tao Taiyuan University of Technology, Department of Mathematics, Associate Professor ; Liu Kai Nanchang University, Department of Mathematics, Associate Professor ; Li Nan University of Jinan, School of Mathematical Sciences, Lecturer ; Zheng Yue-Yang University of Eastern Finland, Department of Physics and Mathematics, Doctor
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: トロピカル・ネバンリンナ理論 / 超離散方程式 / 複素差分方程式 / 複素線型微分方程式 / 超越有理型函数 / Mason's theorem / exponential polynomial / 国際研究者交流

Outline of Final Research Achievements

Laine and I have set up the tropical value distribution theory on a finite open interval by means of an ultradiscrete linear fractional transformation as shift, while some results from function theory have been tropicalized together with Liu. Collaborating with Laine, Korhonen, Heittokangas and Y-Y. Zhang, we have made a progress on complex functional equations, while it was the joint work with Gundersen and Steinmetz to have settled an open problem on the uniqueness of two meromorphic functions sharing five pairs of values affirmatively. Furthermore, we invited Ishizaki, Wen, Li and JーH. Zheng and studied the zero distribution of exponential polynomials, an analogue of Mason's theorem with differential radical and a refinement of the estimate of difference ratio of meromorphic functions each of which provides the best possible result in a certain sense. Finally, Lin and I studied the difference counterpart of sharing values jointly and obtained a sharp result concerning this concept.

Free Research Field

複素解析

Academic Significance and Societal Importance of the Research Achievements

「差分による微分の代替可能性」を調べるべく有理型函数を対象にシフト作用素に関する結果を導き、微分作用素による場合と平行な推論を用いて証明した。また区分的線型な連続関数がmax-plus級数を持つことを利用して用語・推論共に機械的な「翻訳」により応用上有益な複素解析的な場合と殆ど同等な結果を再現した。これらは誤解を恐れずに表現すれば「ジェネリック薬品」的な成果で、更なる進展により学術的意義と社会的意義が更に高まると期待する。