Research on the belonging problem of polynomial automorphisms and related fields

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03219
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: Kuroda Shigeru 東京都立大学, 理学研究科, 教授 (70453032)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 多項式環 / 自己同型群 / 加法群作用 / 不変式環 / 正標数 / 永田自己同型 / Anick自己同型 / イニシャル代数

Outline of Final Research Achievements

Polynomial rings are important subjects in algebra, but their fundamental properties have not yet been fully elucidated, and there are many unsolved problems around polynomial rings. In this study, we have investigated the deep properties of polynomial rings from various perspectives, focusing on automorphisms of polynomial rings, and constructed related theories. We have achieved a wide range of results, including determining the structure of invariant rings for automorphisms in positive characteristic, and discovering a new method for constructing finitely generated subalgebras whose initial algebras are infinite.

Free Research Field

可換環論

Academic Significance and Societal Importance of the Research Achievements

正標数の体上の多項式環の自己同型群の構造の解明は，近年の多項式環論における重要な課題の一つである．今回，不変式環などに関して新たな成果が得られたことは，この方面の研究を推進するうえで大きな助けになる．
また，多項式環に関する基本的な結果は，代数学における基礎理論として重要な役割を果たしており，長期的な視点に立てば，代数学が将来さらなる発展を遂げるための礎になることも期待できる．