An Improved Algorithm For Gr?bner Bases

The theoretical foundation and algorithm research of the Gr?bner basis play an important role in the fields of computational algebraic geometry,automated geometry theorem proving,robotics,algebraic coding theory,cryptography,graph theory,etc.,and the Buchberger algorithm is the core of the Gr?bner basis.The algorithm has calculation inefficiency and redundant calculation when solving the ideal Gr?bner basis.Therefore,many scholars have conducted in-depth research on how to improve the calculation efficiency of this algorithm from multiple directions,and have proposed many new algorithms.Among them,the most effective such as: F4 algorithm,G2 V algorithm,GVW algorithm,etc.This paper studies and analyzes the Gr?bner basis algorithm—GR?BNERNEW2 algorithm and F4 algorithm,and tests with Maple platform examples to find that GR?BNERNEW2 algorithm is calculating Cyclic5,Cyclic6,Gerdt 1,2,3,Katsura-5 and other standard polynomial ideal when the Gr?bner basis of the set is used,the calculation efficiency is not high and the program execution time is long.Therefore,by adding a selection strategy to the classical Gr?bner bases algorithm—GR?BNERNEW2,an improved is presented in this paper.The selection strategy is to choose the criterion pairs with the lowest degree of the least common multiple of the least common multiple of the leading monomials of pairs.The proof of the correctness and termination of GR?BNERNEW2C algorithm was given to ensure the integrity of GR?BNERNEW2C algorithm.The algorithm improves the computational efficiency of Gr?bner basis in dealing with multi-variable polynomial ideals,by avoiding some complex computation.Furthermore,GR?BNERNEW2C algorithm,GR?BNERNEW2 algorithm,F4 algorithm,and Buchberger algorithm is implemented with Maple software.Through specific examples,GR?BNERNEW2C algorithm,F4 algorithm,GR?BNERNEW2 algorithm,and Buchberger algorithm were used to solve Cyclic5,6,Gerdt 1,2,3,Katsura-6,7,Haas3,Eco7,Noon6 and other systems,and a table of the computation time of various algorithms is also given in the paper.Finally,this paper applies the improved GR?BNERNEW2C algorithm to the solution of polynomial equations,the additional relationship simplification problem,and the determination of ideal member problems.It lays a certain foundation for further research on the Gr?bner basis algorithm and promotes the development of various disciplines.