Key points are not available for this paper at this time.
A Comprehensive Grobner system for a parametric ideal I in K(A)X represents the collection of all Grobner bases of the ideals I' in KX obtained as the values of the parameters A vary in K. The recent algorithms for computing them comprehensive Grobner systems consider the corresponding ideal J in KA,X, and are based on stability of Grobner bases of ideals under specializations of the parameters. Starting from a Grobner basis of J, the computation splits recursively depending on the vanishing of the evaluation of some ``coefficients'' in KA. In this paper, taking inspiration from the algorithm described by Nabeshima, we create a new iterative algorithm to compute comprehensive Grobner systems. We show how we keep track of the sub-cases to be considered, and how we avoid some redundant computation branches using ``comparatively-cheap'' ideal-membership tests, instead of radical-membership tests.
Bigatti et al. (Sat,) studied this question.