Algebraic methods for determining integrability of discrete equations
| Project/Area Number |
18K13438
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MASE Takafumi 東京大学, 大学院数理科学研究科, 助教 (80780105)
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 可積分系 / 離散可積分系 / 代数的エントロピー / Laurent現象 / 可積分性判定 |
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| Outline of Final Research Achievements |
I studied the integrability of discrete equations by algebraic methods. First, I studied how the choice of an initial value problem of a discrete equation on a multi-dimensional lattice affects its degree growth. I formulated the conditions that a domain must satisfy for integrability. Next, I studied general properties of lattice equations with the Laurent property. I proved that if considered as a set, the Laurent property, the irreducibility and the coprimeness are independent of the choice a domain. Moreover, I studied the method for computing degree growth from singularity pattern. I tried to extend the method to the multi-dimensional case, and I confirmed that the method indeed gives the correct degree growth for several equations. I also studied discrete integrable systems that do not pass the singularity confinement test.
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| Academic Significance and Societal Importance of the Research Achievements |
偏差分方程式や高階の常差分方程式は様々な分野で出現するが、これらの、特に偏差分方程式の可積分性判定について、わかっていることは非常に少ない。今回、領域が満たすべき条件を一般的に定式化したことで、どのような初期値問題を考えるべきか明確にすることができた。また、特異点パターンから次数増大を求める手法を多次元格子の場合に拡張することができたが、これにより、広いクラスの偏差分方程式に対して、次数増大が簡単に予想できるようになった。これは将来に向けた第一歩であり、将来的にこの手法の厳密性が保証されれば、これは次数増大の計算手法として確立し、格子方程式の可積分判定はかなり容易になるだろう。
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Report
(6 results)
Research Products
(15 results)
