| Project/Area Number |
18K03389
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Review Section |
Basic Section 12030:Basic mathematics-related
|
|---|
| Research Institution | University of Tsukuba |
|---|
Principal Investigator |
SASAKI Tateaki 筑波大学, 数理物質系(名誉教授), 名誉教授 (80087436)
|
|---|
| Project Period (FY) |
2018-04-01 – 2024-03-31
|
| Project Status |
Completed (Fiscal Year 2023)
|
| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
|
|---|
| Keywords | 多変数多項式系の変数消去 / 終結式 / 終結式の余計因子 / グレブナー基底 / グレブナー基底法 / 多項式イデアル / イデアルの最低元 / Buchbergerの算法 / 多変数多項式の変数消去 / 多項式剰余列法 / イデアルの生成元係数 / 終結式の余計因子除去 / 多変数終結式法 / 多変数主項消去法 / イデアル最小元の早期計算 / 多項式系の三角化と四角化 / 終結式中の余計因子と除去 / 剰余列法(終結式法) / グレブナー基底の最小元 / イデアル要素の基底係数 / 終結式法とグレブナー基底法の融合 / 多項式系の変数消去 / 剰余列法（終結式法） / 辞書式順序のグレブナー基底 / 消去イデアルの最低元 / 多項式イデアルの最低元 / グレブナー基底計算の高速化 / 多項式剰余列法（終結式法） / 消去イデアルの最小元 / 健康な多変数多項式系 / 剰余列の三角化と四角化 / 多変数多項式系の主変数消去 / 多変数多項式剰余列の最終元 / 多変数多項式系のグレブナー基底 / 消去イデアルの最小元の応用 / 剰余列と余因子列 / 臨界的消去法 |
|---|
| Outline of Final Research Achievements |
As for variable elimination of polynomial systems, we have now two methods. The resultant method can eliminate variables quite fast but the result contains very big extraneous factors, while the Groebner basis (G-base) method gives a complete result but it is very slow.
As for two polynomial system {G,H}, we proved that if we compute the resultant R = res(G,H) and A and B s.t. AG + BH = R, we can remove the extraneous factor of R fully by using ＧＣＤ (Greatest Common Divisor) for A and B. For (m+1)-polynomial system, with m>2, we obtain m resultants by eliminating variables by changing their order, then ＧＣＤ of the resultants is a small multiple of the lowest order element of the ideal. We have also found several methods of computing small multiples.
Thirdly, we developed a method of computing small multiples of G-basis elements from the elements of polynomial remainder sequence efficiently.
|
| Academic Significance and Societal Importance of the Research Achievements |
The bone of Buchberger's algorithm for Groebner basis computation has been almost unchanged more than 60 years, and we had no method for extraneous factor removal for resultants.
This research gave solutions for these many-years unsolved problems, although they must be revised still more.
|
