Abstract
A new efficient algorithm for computing a comprehensive Gröbnersystem of a parametric polynomial ideal over k[U][X] is presented. This algorithm generates fewer branches (segments) compared to previously proposed algorithms including Suzuki and Sato’s algorithm as well as Nabeshima’s algorithm. As a result, the algorithm is able to compute comprehensive Gröbnersystems 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 Gröbnerbasis of an ideal over k[U,X] before performing any branches based on parametric constraints. The proposed algorithm exploits the result that along any branch in a tree corresponding to a comprehensive Gröbnersystem, 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 Singular, and experimented with a number of examples from different applications. Its performance (the number of branches and execution time) has been compared with several other existing algorithms. A number of heuristics and efficient checks have been incorporated into the Magma implementation, especially in the case when the ideal of parametric constraints is 0-dimensional. The algorithm has been successfully used to solve a special case of the famous P3P problem from computer vision.
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