Study of innovative speeding-up of main-variables elimination of multivariate polynomial systems

2023 Fiscal Year Final Research Report

• 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
• 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

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.

Free Research Field

Computer algebra

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.