Improvement Of GVW Algorithm And Its Application In Solving Algebraic Equations

The Groebner basis is a very powerful tool in algebraic geometry,and it is involved in commutative algebra and many other scientific and engineering problems.The concept of Groebner basis was introduced by Buchberger in his doctoral thesis,in which Buchberger also introduced relevant theories of Groebner basis and the algorithm used to calculate Groebner basis(called Buchberger algorithm).After that,he proposed two criteria(called Buchberger first criterion and Buchberger second criterion)to improve this algorithm.Since then,many mathematicians have worked to find more efficient ways to calculate the Groebner basis.Among the subsequent improvements of Buchberger algorithm,Faug`ere’s F4 algorithm and F5 algorithm proposed successively as well as the GVW algorithm proposed by Gao et al.(Gao S,Volny IV F,Wang M.)have been widely used,among which F5 algorithm and GVW algorithm are Groebner based algorithm based on signature.And GVW algorithm can calculate the Groebner basis of ideal I and syzygy module at the same time,so GVW algorithm is one of the most powerful tools for solving Groebner basis.In addition,the die plays an important role in the free resolution of homology algebra,so this algorithm has been applied and improved since it was put forward.Observe the GVW algorithm and find that in every loop,we need to compute the J-pairs generated by all pairs first,and then use the criterion in the GVW algorithm to judge whether these J-pairs are useful,for the useful J-pairs,regulartop-reduction should be done as much as possible.If more useless J-pairs can be identified in each step of the algorithm cycle,the efficiency of the algorithm can be further improved.For this reason,Zheng et al.(Zheng L,Li D,Liu J)proposed the concept of factor based on the criteria of the original GVW algorithm and applied it to the GVW algorithm.However,it is easy to observe that the output results of the algorithm improved by Zheng et al.are not consistent with the approximate Groebner basis,so we can further improve the algorithm.The main content of this paper is that Zheng et al.named the rules used to determine whether J-pairs are useful by factors into factor criterion,presented them in the form of a theorem,and sorted out and improved its proving process.Then,we further improve the GVW algorithm on this basis,so that the final output result of the algorithm is a reduced Groebner basis.To this end,we also need to prove that the output result of the improved algorithm is a reduced Groebner basis.Then,the improved algorithm is applied to solve multiple algebraic equations.By comparing the calculation process and output results of the improved algorithm before and after application,the superiority of the improved algorithm is realized.Finally,Maple software was used to verify the accuracy of the calculation results of the examples in this paper,and Maple program was used to compare the effect of GVW algorithm before and after applying the factor criterion.