Structural analysis of the automorphism group of a polynomial ring and its application

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04826
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: Kuroda Shigeru 首都大学東京, 理学研究科, 教授 (70453032)
• Research Collaborator: Kojima Hideo 新潟大学, 大学院自然科学研究科, 教授 (90332824) ; Tanimoto Ryuji 静岡大学, 教育学部, 准教授 (20547062)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 多項式環 / 自己同型写像 / 安定余順自己同型 / 加法群作用 / ヒルベルトの第14問題

Outline of Final Research Achievements

A polynomial is an indispensable concept in mathematics, and the ring formed by them is a basic object in modern algebra. However, there remain various difficult open problems concerning polynomial rings, which have been studied worldwide. In the study of such problems, automorphisms of a polynomial ring and the ring formed by them play important roles. In this research, we investigated subgroups of the automorphism group of a polynomial ring and related objects, and obtained various new information about them. We also succeeded in constructing a new counterexample to Hilbert's fourteenth problem by using the knowledge of polynomial automorphisms.

Free Research Field

可換環論

Academic Significance and Societal Importance of the Research Achievements

ヒルベルトの基底定理をはじめ，多項式環に関する基本的な結果は，現代代数学において重要な役割を果たしている．多項式環の基本的な性質については未だ不明な点が多く，それらを解明することは，将来的に代数学の発展に多大な貢献をするはずである．多項式環の自己同型や自己同型群は，多項式環を理解するうえで欠かせないものであり，本研究で得られた成果は，今後の多項式環研究で重要な意味を持つと考えられる．