| Project/Area Number |
11640176
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Tokyo Woman's Christian University |
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Principal Investigator |
OAKU Toshinori Tokyo Woman's Christian University, Dept. of Mathematics, Professor, 文理学部, 教授 (60152039)
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| Co-Investigator(Kenkyū-buntansha) |
KONDO Takeshi Tokyo Woman's Christian University, Dept. of Mathematics, Professor, 文理学部, 教授 (20012338)
KOBAYASHI Kazuaki Tokyo Woman's Christian University, Dept. of Mathematics, Professor, 文理学部, 教授 (50031323)
MIYACHI Akihiko Tokyo Woman's Christian University, Dept. of Mathematics, Professor, 文理学部, 教授 (60107696)
YAMASHIMA Seiho Tokyo Woman's Christian University, Dept. of Mathematics, Associate Professor, 文理学部, 助教授 (80086347)
SHINOHARA Masahiko Tokyo Woman's Christian University, Dept. of Mathematics, Professor, 文理学部, 教授 (70086346)
永山 操 東京女子大学, 文理学部, 講師 (30237557)
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| Project Period (FY) |
1999 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2000: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1999: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | D-module / linear partial differential equation / algorithm / Groebner base / symbolic computation / algebraic analysis / 自由分解 |
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| Research Abstract |
D-modules stand for modules over the ring of differential operators, which corresponds to systems of linear partial differential equations. The theory of D-modules has been developed since 1970's mainly by M.Sato, M.Kashiwara and T.Kawai. However, there was no systematic study on actual computation for D-modules. By the collaboration With N.Takayama, I have found algorithms for D-modules applying Groebner basis techniques to D-modules. In particular, we have found algorithms for computing the cohomology groups associated with the inverse image (restriction) and the direct image (integration) of a D- module. One of the essential ingredients in these algorithsms is the computation of free resolutions for D-modules which are adapted to a filtration (or a weight vector). However free resolutions we obtained were often too big to complete the computation.
In order to overcome this bottleneck, we have introduced the notion of minimal free resolution that is adapted to a filtration and have shown that such a minial free resolution is computable. This minimal resolution algorithm enables us to compute, e.g., algebraic de Rham cohomology groups and algebraic local cohomology groups more effciently. N.Takayama has implemented this algorithm in his computer algebra system Kan and has opened it to the public via internet.
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