Generic Gröbner basis of a parametric ideal and its application to a comprehensive Gröbner system
Katsusuke Nabeshima · Applicable Algebra in Engineering Communication and Computing · 2023
Abstract A relationship between a parametric ideal and its generic Gröbner basis is discussed, and a new stability condition of a Gröbner basis is given. It is shown that the ideal quotient of the parametric ideal by the ideal generated using the generic Gröbner basis provides the stability condition. Further, the stability condition is utilized to compute a comprehensive Gröbner system, and hence as an application, we obtain a new algorithm for computing comprehensive Gröbner systems. The proposed algorithm has been implemented in the computer algebra system Risa/Asir and experimented with a number of examples. Its performance (execution timings) has been compared with the Kapur-Sun-Wang’s algorithm for computing comprehensive Gröbner systems.