2022 Fiscal Year Final Research Report
Research of difference equations in the complex domains and its applications
| Project/Area Number |
20K03658
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | The Open University of Japan |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
藤解 和也 金沢大学, 電子情報通信学系, 教授 (30260558)
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| Project Period (FY) |
2020-04-01 – 2023-03-31
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| Keywords | Differential equations / Exponential polynomials / Functional equations / Nevanlinna theory / Difference equations / Wiman-Valiron theory / Growth of order / Stothers-Mason theorem |
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| Outline of Final Research Achievements |
By means of the Nevanlinna theory, we obtained several results on the existence of meromorphic solutions to linear differential equations and linear difference equations, in which we discussed the value distribution and the order of growth of meromorphic solutions. In particular, we investigated the properties on the value distribution of solutions in connection with the those of coefficients when exponential polynomials are included in the coefficients. We also have been concerned with some open questions of the Fermat type functional equations and difference analogues of them, and obtained partial answers to these questions and gave alternative proofs of some known results. Further, we introduced the idea of the difference radical and proved the difference analogues of the Stothers-Mason theorem.
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| Free Research Field |
解析学
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| Academic Significance and Societal Importance of the Research Achievements |
20世紀前半に確立されたNevanlinna理論は、線型・非線型を問わず複素領域での微分方程式の有理型函数解を調べることに対して有効である。しかしながら、差分方程式やFermat型方程式などの函数方程式を取り扱うためには、Nevanlinna理論のそれぞれの方程式に対応する新たな展開が必要である。本研究は、基礎を支える理論構築と応用面の新技法提案からなる上昇螺旋を描き、自然科学における基礎研究の重要さを記述していると期待する。
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