Arithmetic of division polynomials and orthogonal polynomials
| Project/Area Number |
17K05168
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Nagoya Institute of Technology |
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Principal Investigator |
Yamagishi Masakazu 名古屋工業大学, 工学(系)研究科(研究院), 教授 (40270996)
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| Co-Investigator(Kenkyū-buntansha) |
水澤 靖 名古屋工業大学, 工学(系)研究科(研究院), 教授 (60453817)
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| Project Period (FY) |
2017-04-01 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2020)
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| Budget Amount *help |
¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | 等分多項式 / チェビシェフ多項式 / ヤコビ楕円関数 / 終結式と判別式 / 線形符号 / 重み多項式 / 形式群 / リュカ数列 / ディクソン多項式 / 終結式 / 判別式 / 楕円関数 / 直交多項式 |
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| Outline of Final Research Achievements |
Division polynomials are important subjects in algebraic number theory. On the other hand, orthogonal polynomials often appear in graph theory and in combinatorics; in particular, Chebyshev polynomials have wide-ranging applications. By regarding Chebyshev polynomials as division polynomials, we are able to investigate their properties by arithmetic methods and to apply Chebyshev polynomials to various mathematical problems. In this study, we applied this point of view to other kinds of division polynomials. The main results are the determination of resultants of division polynomials of Jacobi elliptic funcions, and an application of division polynomials of formal groups. Also, as initially unexpected results, we gave answers to some problems in coding theory by using Chebyshev polynomials.
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| Academic Significance and Societal Importance of the Research Achievements |
各種多項式系列の終結式は古くから計算され、代数的整数論を始め各分野で応用されている。最近では数論力学系に関連して、ワイエルシュトラス楕円関数の等分多項式の終結式が計算された（Harry Schmidt, 2015）。その系列にヤコビ楕円関数の等分多項式を加えることができたことは学術的に意義があるといえる。虚2次体の整数環の単生性への応用が期待できる。符号理論に関する本研究の成果は理論的なものであり、実在の線形符号に直接関わるものではないが、将来的には符号のゼータ関数について新しい知見を与えることが期待される。
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Report
(5 results)
Research Products
(8 results)
