Efficiency improvements concerning real computational algebraic methods and their application in the mathematical sciences

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K19745
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 60010:Theory of informatics-related
• Research Institution: Kyushu University
• Principal Investigator: Fukasaku Ryoya 九州大学, 数理学研究院, 助教 (40778924)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 包括的グレブナー基底系 / ホップ分岐

Outline of Final Research Achievements

We have achieved certain results in this research project by proposing an algebraic method related to Hopf bifurcations with fixed multiplicities, as well as a simplified representation of comprehensive Groebner systems and other methods. In particular, the multiplicities of Hop bifurcations is a concept that serves as an upper bound on the number of limit cycles. It is significant that we were able to propose a computational algebra method that is relevant to dynamical systems. In addition, the simplicities of the representations of comprehensive Groebner systems has a strong influence on the computation time and memory used for real quantifier eliminations. Since computational algebra methods tend to require a large amount of computation time and memory, the simplicity of the representation of comprehensive Groebner systems is one of important topics related to computational efficiency.

Free Research Field

計算代数

Academic Significance and Societal Importance of the Research Achievements

計算代数手法は厳密な計算やパラメータを記号的に使うような計算を行うことができる。一方で、数値計算手法などと比べて膨大な計算資源（計算時間・計算メモリなど）を要求し易いような傾向を持っている。本研究では、厳密な計算やパラメータを記号的に使うような計算が可能であるというメリットを数理科学分野に活用してきた。また、膨大な計算資源が要求され易い傾向を持つというデメリットを、計算の効率化などを目指すことで、解消しようとする研究であると考えている。そして、今後、多くの学術的問題・社会的問題に計算代数手法を用いることを目指しているという点に学術的意義や社会的意義を持っていると考える。