2022 Fiscal Year Final Research Report
Characterization of polynomials which satisfy local functional equations
| Project/Area Number |
17K05209
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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 | Josai University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2023-03-31
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| Keywords | 局所関数等式 / 概均質ベクトル空間 / ゼータ超関数 / Clifford quartic form / polarization / F-多項式 / Catalecticant / homaloidal多項式 |
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| Outline of Final Research Achievements |
The following research results were obtained on pairs of polynomials satisfying local function equation.
(1) Explicit determination of local function equation associated with polarization of homaloidal polynomials and proof that prehomogeneity is maintained by polarization(2) Observation of polarization of polynomials satisfying local functional equation even if they are non-prehomogeneous homogeneous(3) Observation of prehomogenety of each uniform part of F-polynomials associated with cluster algebra with coefficients and observation of their relationship with projective manifolds(4) Observation of prehomogeneity of a section of Catalecticant determinant(5) Relationship between resultant and generalized resultants and prehomogeneous vector space
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| Free Research Field |
表現論と整数論
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| Academic Significance and Societal Importance of the Research Achievements |
局所関数等式の研究は整数論では保型形式、ゼータ関数などの整数論の研究で重要であり, 今までの流れでは局所関数等式を満たす多項式のタイプはそれらの分野では概均質タイプのみしか扱ってこなかったが, 非概均質タイプまで含めた今回の研究は今までにないタイプの局所関数等式やその背後にある空間の特性を調べたものなので,整数論で、新しい現象の発見につながることが期待できる.表現論の分野では, 概均質ベクトル空間は群がreductiveであるが, 今回の研究では群がSolvableなものも含んでいて局所関数等式が多変数になり, 等質錐に付随するLaplace-Fourier変換との関係など大いに期待される.
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