1998 Fiscal Year Final Research Report Summary
Study on Algorithms for D-Modules
| Project/Area Number |
09640055
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Yokohama City University |
|---|
Principal Investigator |
OAKU Toshonori Yokohama City Univ., Dept. of Mathematical Sciences, Associate Professor, 理学部, 助教授 (60152039)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
FUJII Michihiko Yokohama City Univ., Dept. of Mathematical Sciences, Assistant, 理学部, 助手 (60254231)
FUJII Kazuyuki Yokohama City Univ., Dept. of Mathematical Sciences, Associate Professor, 理学部, 助教授 (00128084)
ASANO Hiroshi Yokohama City Univ., Dept. of Mathematical Sciences, Professor, 理学部, 教授 (00046012)
TAKAYAMA Nobuki Kobe Univ., Dept. of Mathematics, Professor, 理学部, 教授 (30188099)
|
|---|
| Project Period (FY) |
1997 – 1998
|
| Keywords | D-module / linear partial differential equation / algorithm / Groebner base / symbolic computation |
|---|
| Research Abstract |
Our purpose has been to establish algorithms for computing various operations (functors) for D-modules by applying the method of Groebner base to the ring of differential operators and to develop software for doing such computations. Our results are as follows :
1. Algorithms for computing functors of D-modules : We have obtained an algorithm for comoputing the (cohomology groups of) the restriction to an affine subvariety, and the integration along an affine space, of a D-module defined on an affine space. This is done by computing a free resolution adapted to some filtration.
2. Algorithms for holonomic functions : It is difficult to treat various functions by computers. Dealing with the differential equations which a function satisfies rather than the function itself, we obtained algorithms for computing the system of differential equtions for the integration and the product of holonomic functions.
3. An algorithm for de Rham cohomology : As an application of the algorithm for computing the integration, we obtained an algorithm to compute the de Rham cohomology groups of the complement of an algebraic set in the affine space.
4. Development and distribution of the software : A system kan, which N.Takayama has been developing from around 1991, is a program for computing D-modules mainly by Groebner basis computation in the ring of differential operators. We have imple- mented the algorihms mentioned above in kan. The programs are freely distributed from http : //www.math.kobe-u.ac.jp/KAN/ along with source codes.
|
