Improving the DISPGB algorithm using the discriminant ideal

In 1992, V. Weispfenning proved the existence of Comprehensive Gröbner Bases (CGB) and gave an algorithm for computing one. That algorithm was not very efficient and not canonical. Using his suggestions, A. Montes obtained in 2002 a more efficient algorithm (DISPGB) for Discussing Parametric Gröbner Bases. Inspired by its philosophy, V. Weispfenning defined, in 2002, how to obtain a Canonical Comprehensive Gröbner Basis (CCGB) for parametric polynomial ideals, and provided a constructive method. In this paper we use Weispfenning’s CCGB ideas to make substantial improvements on Montes’ DISPGB algorithm. It now includes rewriting of the discussion tree using the discriminant ideal and provides a compact and effective discussion. We also describe the new algorithms in the DPGB library containing the improved DISPGB as well as new routines for checking whether a given basis is a CGB or not, and for obtaining a CGB. Examples and tests are also provided.