2000 Fiscal Year Final Research Report Summary
Study on Algorithms for D-Modules
| 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)
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| Project Period (FY) |
1999 – 2000
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| Keywords | D-module / linear partial differential equation / algorithm / Groebner base / symbolic computation / algebraic analysis |
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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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Research Products
(11 results)
