Searching for integrable systems over finite fields

• Project/Area Number: 15K04904
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Future University-Hakodate
• Principal Investigator: Yura Fumitaka 公立はこだて未来大学, システム情報科学部, 教授 (90404805)
• Project Period (FY): 2015-10-21 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: soliton / finite field / 離散可積分系 / 楕円曲線 / 楕円数列

Outline of Final Research Achievements

In this research, I have mainly constructed the dynamical systems related to the lattice model that take a value on finite fields. The solitonic systems obtained there over finite fields conserves solitary waves, and have polynomial representations, which are novel dynamical systems. The properties of the general solutions of the elliptic sequences and the Somos sequences as a special case are also considered through Hankel determinants. The elliptic sequence is equivalent to the sequence of points on an elliptic curve, and fundamental object for the elliptic curve cryptography and the algebraic geometry. Furthermore, a basic application to cryptography of algebraic systems connected with dynamical systems that support a soliton equation over finite fields are considered.

Academic Significance and Societal Importance of the Research Achievements

まず、有限体上において孤立波を保存するようなソリトン系は従来得られていなかったと思われるため、新規な力学系である。また多項式表示を持つ点は、従来の実数あるいは複素数上の可積分系には見られない大きな特徴であり、離散的な系との比較は可積分系に対する新しい視点となりうる。また、ここに現れる枠組みは平方剰余と関係することから、力学系として新規なモデルを与える可能性だけではなく、代数系を基にした暗号理論への応用が今後期待される。

Research Products

• [Journal Article] 血管新生の数理モデル (2016)
  • Author(s): 間田潤, 松家敬介, 由良文孝, 栗原裕基, 時弘哲治
  • Journal Title: 日本応用数理学会論文誌 Volume: 26(1) Pages: 105-123
  • NAID: 110010042558
  • Peer Reviewed / Open Access
• [Journal Article] 楕円曲線とHankel行列式 (2015)
  • Author(s): 由良文孝
  • Journal Title: 九州大学応用力学研究所講究録 Volume: 26AO-S2 Pages: 163-169
  • Peer Reviewed / Open Access
• [Journal Article] Hankel determinant solution for elliptic sequence (2015)
  • Author(s): Fumitaka Yura
  • Journal Title: Linear Algebra and Its Applications Volume: 484 Pages: 27-45
  • DOI: 10.1016/j.laa.2015.06.016
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] 大域的2次元セルオートマトンのある拡張について (2018)
  • Author(s): 由良 文孝
  • Organizer: 日本応用数理学会