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Algebraic methods for determining integrability of discrete equations

Research Project

Project/Area Number 18K13438
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionThe University of Tokyo

Principal Investigator

MASE Takafumi  東京大学, 大学院数理科学研究科, 助教 (80780105)

Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
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)
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
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.

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。

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (15 results)

All 2023 2021 2020 2019 2018

All Journal Article (12 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 11 results,  Open Access: 7 results) Presentation (3 results) (of which Int'l Joint Research: 3 results)

  • [Journal Article] Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice2021

    • Author(s)
      Kamiya R.、Kanki M.、Mase T.、Tokihiro T.
    • Journal Title

      Journal of Mathematical Physics

      Volume: 62 Issue: 10 Pages: 102701-102701

    • DOI

      10.1063/5.0034581

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Coordination sequences of crystals are of quasi-polynomial type2021

    • Author(s)
      Nakamura Yusuke、Sakamoto Ryotaro、Mase Takafumi、Nakagawa Junichi
    • Journal Title

      Acta Crystallographica Section A Foundations and Advances

      Volume: 77 Issue: 2 Pages: 138-148

    • DOI

      10.1107/s2053273320016769

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Algebraic entropy of a multi-term recurrence of the Hietarinta-Viallet type2020

    • Author(s)
      R. Kamiya, M. Kanki, T.Mase, T. Tokihiro
    • Journal Title

      数理解析研究所講究録別冊

      Volume: B78 Pages: 121-153

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Algebraic entropy computations for lattice equations: why initial value problems do matter2019

    • Author(s)
      J. Hietarinta, T. Mase and R. Willox
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 52 Issue: 49 Pages: 49LT01-49LT01

    • DOI

      10.1088/1751-8121/ab5238

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Singularity confinement as an integrability criterion2019

    • Author(s)
      Mase Takafumi、Willox Ralph、Ramani Alfred、Grammaticos Basil
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 52 Issue: 20 Pages: 205201-205201

    • DOI

      10.1088/1751-8121/ab1433

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Toda type equations over multi-dimensional lattices2018

    • Author(s)
      Kamiya Ryo、Kanki Masataka、Mase Takafumi、Okubo Naoto、Tokihiro Tetsuji
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 51 Issue: 36 Pages: 364002-364002

    • DOI

      10.1088/1751-8121/aad375

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Nonlinear forms of coprimeness preserving extensions to the Somos-4 recurrence and the two-dimensional Toda lattice equation?investigation into their extended Laurent properties2018

    • Author(s)
      Kamiya Ryo、Kanki Masataka、Mase Takafumi、Tokihiro Tetsuji
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 51 Issue: 35 Pages: 355202-355202

    • DOI

      10.1088/1751-8121/aad074

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Studies on spaces of initial conditions for non-autonomous mappings of the plane2018

    • Author(s)
      Mase Takafumi
    • Journal Title

      Journal of Integrable Systems

      Volume: 3 Issue: 1

    • DOI

      10.1093/integr/xyy010

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] On the Coprimeness Property of Discrete Systems without the Irreducibility Condition2018

    • Author(s)
      Kanki Masataka、Mase Takafumi、Tokihiro Tetsuji
    • Journal Title

      Symmetry, Integrability and Geometry: Methods and Applications

      Volume: 14 Pages: 065-065

    • DOI

      10.3842/sigma.2018.065

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Calculating the algebraic entropy of mappings with unconfined singularities2018

    • Author(s)
      Ramani A、Grammaticos B、Willox R、Mase T、Satsuma J
    • Journal Title

      Journal of Integrable Systems

      Volume: 3 Issue: 1

    • DOI

      10.1093/integr/xyy006

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Integrable mappings and the notion of anticonfinement2018

    • Author(s)
      Mase T、Willox R、Ramani A、Grammaticos B
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 51 Issue: 26 Pages: 265201-265201

    • DOI

      10.1088/1751-8121/aac578

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] 多次元格子上の擬似可積分系2018

    • Author(s)
      神吉雅崇、時弘哲治、間瀬崇史
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 2071 Pages: 43-64

    • Related Report
      2018 Research-status Report
    • Open Access
  • [Presentation] The Laurent property, irreducibility and coprimeness of non-integrable partial difference equations2023

    • Author(s)
      T. Mase
    • Organizer
      Pure maths colloquium talks
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Integrability tests for lattice equations - or why initial value problems do matter2019

    • Author(s)
      T. Mase
    • Organizer
      Integrable Systems 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Dynamical degrees and singularity patterns2018

    • Author(s)
      T. Mase, R. Willox, A. Ramani, B. Grammaticos
    • Organizer
      International Conference on Symmetry and Integrability in Difference Equations
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research

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Published: 2018-04-23   Modified: 2024-01-30  

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