A new algorithm for computing comprehensive Gröbner systems

A new algorithm for computing a comprehensive Gröbner system of a parametric polynomial ideal over k[U][X] is presented. This algorithm generates fewer branches (segments) compared to Suzuki and Sato's algorithm as well as Nabeshima's algorithm, resulting in considerable efficiency. As a result, the algorithm is able to compute comprehensive Gröbner systems of parametric polynomial ideals arising from applications which have been beyond the reach of other well known algorithms. The starting point of the new algorithm is Weispfenning's algorithm with a key insight by Suzuki and Sato who proposed computing first a Gröbner basis of an ideal over k[U,X] before performing any branches based on parametric constraints. Based on Kalkbrener's results about stability and specialization of Gröbner basis of ideals, the proposed algorithm exploits the result that along any branch in a tree corresponding to a comprehensive Gröbner system, it is only necessary to consider one polynomial for each nondivisible leading power product in k(U)[X] with the condition that the product of their leading coefficients is not 0; other branches correspond to the cases where this product is 0. In addition, for dealing with a disequality parametric constraint, a probabilistic check is employed for radical membership test of an ideal of parametric constraints. This is in contrast to a general expensive check based on Rabinovitch's trick using a new variable as in Nabeshima's algorithm. The proposed algorithm has been implemented in Magma and experimented with a number of examples from different applications. Its performance (vis a vie number of branches and execution timings) has been compared with the Suzuki-Sato's algorithm and Nabeshima's speed-up algorithm. The algorithm has been successfully used to solve the famous P3P problem from computer vision.