2007 Fiscal Year Final Research Report Summary
Study on algorithms for D-modules
| Project/Area Number |
16540172
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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 Toshiimori Tokyo Woman's Christian University, Department of Mathematics, Professor (60152039)
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| Co-Investigator(Kenkyū-buntansha) |
MIYACHI Akihiko Tokyo Woman's Christian University, Department of Mathematics, Professor (60107696)
KOBAYASHI Kazuaki Tokyo Woman's Christian University, Department of Mathematics, Professor (50031323)
OHYAMA Yoshiyuki Tokyo Woman's Christian University, Department of Mathematics, Professor (80223981)
SHINOHARA Masahiko Tokyo Woman's Christian University, Department of Mathematics, Professor (70086346)
KODATE Takako Tokyo Woman's Christian University, Department of Mathematics, Lecturer (90317826)
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| Project Period (FY) |
2004 – 2007
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| Keywords | D-module / partial differential equations / algorithm / Groebner base / regular singularity / b-function |
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| Research Abstract |
1. In general, a D-module, or a system of linear partial differential equations is said to be regular specializable along a submanifold if it has regular singularities along the submanifold. Then the formal power series solutions with respect to the variables transversal to the submanifold converge as was proved by M. Kashiwara and T. Oshima in 1970's. However, for a given D-module, it is not easy to determine if it is regular specializable along a submanifold. By using the homogenization technique and the division algorithm introduced by the previous work of mine collaborated with M. Granger and N. Takayama, I have got a complete algorithm to decide if a given algebraic D-modules is regular specializable along a linear submanifold. At the same time, one can compute what is called the regular b-function of the D-module.
I also made programs for the above algorithms using a computer algebra system KAN developed by N. Takayama. In particular, I applied these programs to A-hypergeometric systems defined by Gelfand‐Kapranov-Zelevinski. As a result, I conjectured that A-hypergeometric systems are always regular specializable along the origin. I proved this conjecture also by using the homogenization technique. This implies that the formal power series solutions of an A-hypergeometric system always converge around the origin.
2. For several polynomials, one can associated a polynomial ideal called the Bernstein-Sato ideal. Collaborating with R.Bahloul, I obtained an algorithm to compute the Bernstein-Sato ideal. I made a program for that algorithm by using a computer algebra system Risa/Asir developed by M. Noro. With this program, we computed several non-trivial examples of Bernstein-Sato ideals completely. For examples, we decided the generators of the Bernstein-Sato ideal which was previously studied by Briancon and Maynadier and was shown to be non-principal.
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