2013 Fiscal Year Final Research Report
Algorithm for efficiently computing a Groebner basis with high probability
| Project/Area Number |
23654035
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kobe University |
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Principal Investigator |
NORO Masayuki 神戸大学, 理学(系)研究科(研究院), 教授 (50332755)
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| Project Period (FY) |
2011 – 2013
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| Keywords | 応用数学 / 計算代数 / グレブナー基底 / モジュラー計算 |
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| Research Abstract |
We found that the cost of integer-rational number conversion in the procedure for
computing Groebner bases over the rationals by combining modular Groebner bases by Chinese remainder theorem can be reduced by converting all the polynomials with the same degree together.
In order to show the correctness of a Groebner basis candidate, we developed an algorithm which computes an exact generating relation for each element in the candidate. We first compute a generating relation over a finite field. Then we replace the coefficients with variables to obtain a huge system of linear equations. By using the information obtained by solving the system over the finite field, we can reduce the size of the system and the solution is made unique. Then we can apply Hensel lifting for solving the system and we can efficiently solve the system.
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