2003 Fiscal Year Final Research Report Summary
Study on Algorithms for D-Modules.
| Project/Area Number |
13640192
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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, Professor, 文理学部, 教授 (60152039)
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| Co-Investigator(Kenkyū-buntansha) |
SHINOHARA Masahiko Tokyo Woman's Christian University, Dept. of Mathematics, Professor, 文理学部, 教授 (70086346)
MIYACHI Akihiko Tokyo Woman's Christian University, Dept. of Mathematics, Professor, 文理学部, 教授 (60107696)
KOBAYASHI Kazuaki Tokyo Woman's Christian University, Dept. of Mathematics, Professor, 文理学部, 教授 (50031323)
YAMASHITA Shigeho Tokyo Woman's Christian University, Dept. of Mathematics, Associate Professor, 文理学部, 助教授 (80086347)
OYHAMA Yoshiyuki Tokyo Woman's Christian University, Dept. of Mathematics, Associate Professor, 文理学部, 教授 (80223981)
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| Project Period (FY) |
2001 – 2003
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| Keywords | D-module / lihear partial differential equation / algorithm / Groebner base / free resolution / minimal resolution / symbolic computation / division |
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| Research Abstract |
1. Collaborating with Professor Nobuki Takayama of Kobe University, I defined the notion of minimal (filtered) free resolution for a module over the homogenized ring of the Weyl algebra (i.e., the ring of differential operators with polynomial coefficients). We also introduced an efficient algorithm for computing a minimal free resolution of a module over the homogenized Weyl algebra. This algorithm was implemented in software Nan.
On the other -hand, using the homogenization of the ring of analytic differential operators with respect to the order filtration, I defined the notion of minimal filtered free resolution for a module over this homogenized ring of analytic differential operators with Professor M. Granger of Angers University, France. We proved that such a minimal filtered free resolution exists uniquely up to isomorphism of complexes. As an application, we introduced a set of numerical invariants of analytic hypersurface singularities.
2. M. Granger and I found a division algorithm in a finite free module over the homogenized ring of analytic differential operators which is generated by operators with polynomial coefficients. This is an extension to D-modules of a celebrated tangent cone algorithm of T. Mora for power series. Takayama implemented this algorithm in Nan. Being able to work with' any monomial ordering compatible with the module-structure, this is one of the most general division algorithms for D-modules.
3. N. Takayama, Y. Shiraki and I studied the method of numerical integration for special functions with parameters by using algorithm for D-modules : We showed that for some examples, this, new method is more efficient than the classical 'method of numerical integration.
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