Formulations of the tropical Nevanlinna-Cartan theory and their returns for complex analytic methods

2016 Fiscal Year Final Research Report

• Project/Area Number: 25400131
• 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, 電子情報学系, 教授 (30260558)
• Co-Investigator(Renkei-kenkyūsha): SHIMOMURA Shun 慶応義塾大学, 理工学部, 教授 (00154328) ; ISHIZAKI Katsuya 放送大学, 教養部, 教授 (60202991)
• Research Collaborator: LAINE Ilpo 東フィンランド大学, 物理数学科, 名誉教授 ; KORHONEN Risto 東フィンランド大学, 物理数学科, 教授 ; HEITTOKANGAS Janne 東フィンランド大学, 物理数学科, 准教授
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: Nevanlinna理論 / 有理型函数 / 値分布論 / 微分方程式 / トロピカル数学 / max-plus 代数 / 超離散方程式 / 国際交流者研究

Outline of Final Research Achievements

There are known many applications of analytic functions not only to mathematics but also to a broad range of fields in science and engineering, where the essential role is played by power series expansion. Actually, most of them are functions which permit meromorphic continuation over the whole complex plane and solve some distinctive equations such as differential equations for exponential or elliptic functions. It was 90 years ago when Rolf Nevanlinna established his theory on value distribution of the tanscendental meromorphic functions. Our study observed the question whether such contributions can be done only with complex analysis or there is a possible replacement of the role. It is our main result that we have formulated the tropical analogues of the complex analytic counterparts as a sort of dictionary in a satisfactly fashion for our purpose. In fact, it is found that there is a chance for similar applications by some tropical entire functions with max-plus series expansion.

Free Research Field

複素解析