Perfect algebraic independence properties over non-Archimedean valuation fields
| Project/Area Number |
15K04792
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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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| Research Field |
Algebra
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| Research Institution | Keio University |
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Principal Investigator |
Tanaka Taka-aki 慶應義塾大学, 理工学部(矢上), 准教授 (60306850)
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | 代数的独立性 / Mahler関数 / p進数 / 正標数 / 超越数 / 行列値関数 / 行列環 / p進絶対値 / 群作用 / 代数的独立 / Hecke-Mahler級数 |
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| Outline of Final Research Achievements |
In this project the research representative constructed certain functions having ultimate algebraic independence properties especially over typical non-Archimedean valuation fields. First, he obtained the infinite algebraically independent sets consisting of numbers which can be regarded both as real and as p-adic for a finite number of primes p. Then he established the base for constructing functions having perfect algebraic independence properties over p-adic number fields. Over function fields of positive characteristic, the research representative constructed, using Mahler functions of several variables, the functions having differential perfect algebraic independence properties. Finally, he extended the concept of transcendence and algebraic independence to matrix rings as applications of the functions having differential perfect algebraic independence properties.
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| Academic Significance and Societal Importance of the Research Achievements |
超越数の間の構造を決定することが超越数論の究極の目標であるが、現状ではこの目標は遥か先にある。超越数の構造決定の前段階として、無限集合でその任意の有限部分集合が有理数体上で代数的独立な超越数から成るものの量産が重要である。なぜなら、そのような超越数たちを有理数体に添加して得られる拡大体を最も効率良く最大化できるからである。従って、この目的を単独の関数によって達成できる完全代数的独立性および微分完全代数的独立を有する関数の構成は学術的に意義深い。また、本補助金による社会的貢献の一環として、国際研究集会を主宰し多くの参加者を得て２国間のみならず多国間の国際共同研究の発展に寄与したことが挙げられる。
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Report
(6 results)
Research Products
(9 results)
