2018 Fiscal Year Final Research Report
Optimization of the experimental design by the determinant formula of the relative class numbers of cyclotomic field
| Project/Area Number |
16K12395
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Mathematical informatics
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| Research Institution | Kanazawa Institute of Technology |
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Principal Investigator |
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| Research Collaborator |
Hirabayashi Mikihito
Fujii Satoshi
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | 円分体 / 相対類数 / 行列式 / 実験計画法 / 組み合わせ最適化 / ランダム行列 |
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| Outline of Final Research Achievements |
With regard to the new relative class number determinant formula, we have constructed a matrix with 0, ± 1 components, and a matrix combining the Demjanenko matrix and the secondary index.
We calculated the relative class numbers of 2^31th cyclotomic field etc. This corresponds to the fact that the determinant of about 500 million-order matrix is calculated, and it has been demonstrated that it is possible to calculate objects that are very difficult to calculate directly as the determinant.
We have formulated the prediction of the asymptotic behavior of the Demjanenko matrix and the asymptotic behavior of the D-efficiency in the case of prime p-circles, the former is shown without any special assumptions, the latter under Kummer conjectures.
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| Free Research Field |
代数学, 整数論，計算数論
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| Academic Significance and Societal Importance of the Research Achievements |
一般に，組み合わせ最適化に関連した計算の計算量はNP 困難問題であることが多い．本研究で提案する相対類数計算はO(n^2)の多項式時間で実行可能であり，その計算量の評価も行っている．従って相対類数を表現する±1成分の行列式の値も同計算量で求まり，その結果D-optimal design の近似解も多項式時間で得られる．純粋な整数論的対象である円分体の相対類数およびその行列式公式と，具体的な応用の計算の対象であるD-optimal design の間をつなぐ，最初の一歩を本研究では踏み出した．
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