Extended \(F_5\) criteria
Let \(R = K[x_1,\dots,x_n]\) be a polynomial ring with coefficients in a computable field \(K\) and let \(I\) be the ideal generated by the polynomials \(f_1,\dots,f_k\). \textit{J.-C. Faugère} [ISSAC 2002. Proceedings of the 2002 international symposium on symbolic and algebraic computation, Lille, France, July 07--10, 2002. New York, NY: ACM Press. 75--83 (2002; Zbl 1072.68664)] gave an optimal criterion to compute the Gröbner basis of \(I\) with respect to a term ordering \(<\) and projected a very efficient algorithm called \(F_5\). \(F_5\) is incremental, that is to compute the Gröbner basis of \(I = \langle f_1,\dots,f_k\rangle\), \(F_5\) computes successively the Gröbner basis of the ideals \(\langle f_k\rangle\), \(\langle f_{k-1},f_k\rangle\), \dots, \(\langle f_1,\dots,f_k\rangle\), and its efficiency is due to \(F_5\) criteria, for which the algorithm assigns to any polynomial of \(R\) a signature, that is an element of the module \(R^k\), and that explain how to take advantage of this additionally structure. Hence the quickness of the computation can be affected by the order of the generators and by the module ordering defined on \(R^k\). In this paper the authors modify \(F_5\) criteria considering module orderings coinciding with the term ordering \(<\) of \(R\) on any restriction to a summand of the module \(R^k\). These new criteria lead to an algorithm no more incremental and so not depending on the order of the generators of the ideal. The authors conduct tests on some challenging examples to compare their new proposal with \(F_5\). From this experimental comparison, the new algorithm turns out to be more stable and a little faster than \(F_5\).
scientific article; zbMATH DE number 5817626