In this paper, we improve the GVWDisPGB algorithm for computing comprehensive Gröbner systems developed by Hashemi et al. (Comput Sci J Mold 25(3(75)):278–302, 2017). The GVWDisPGB algorithm is equipped with an incremental structure using the GVW algorithm (Gao et al. in Math Comput 85(297):449–465, 2016), which takes advantage of the F \(_5\) criteria (Faugère in Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ISSAC 2002, Lille, France, July 07–10, 2002, pp. 75–83. ACM Press, New York, NY, 2002) to remove superfluous reductions. While constructing the Gröbner basis of a given ideal in the polynomial ring of variables, this algorithm creates new branches in the parameters space to cover all possible parameter values. To improve this algorithm, we use parametric linear algebra techniques for the initial (not complete) partitioning of parameters space. More precisely, we install a minor modified GES algorithm (Darmian et al. in J Symb Comput 82:38–56, 2017) on the GVWDisPGB algorithm to compute a Gaussian elimination system of parametric matrices (corresponding parametric linear ideal). We show that there exist many examples such that our algorithm, the so-called GES-GVW-CGS algorithm, is faster than the GVWDisPGB algorithm because of constructing fewer branches and consequently having less complexity. All mentioned algorithms have been implemented in Maple, and their efficiency has experimented on a diverse set of benchmark polynomials.
