Solving parametric algebraic systems
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
On the stability of Groübner bases under specializations
Journal of Symbolic Computation
A new algorithm for discussing Gröbner bases with parameters
Journal of Symbolic Computation
Complete Solution Classification for the Perspective-Three-Point Problem
IEEE Transactions on Pattern Analysis and Machine Intelligence
An alternative approach to comprehensive Gröbner bases
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Canonical comprehensive Gröbner bases
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
A speed-up of the algorithm for computing comprehensive Gröbner systems
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Gröbner bases for families of affine or projective schemes
Journal of Symbolic Computation
Minimal canonical comprehensive Gröbner systems
Journal of Symbolic Computation
Automatic discovery of geometry theorems using minimal canonical comprehensive Gröbner systems
ADG'06 Proceedings of the 6th international conference on Automated deduction in geometry
A new algorithm for computing comprehensive Gröbner systems
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Gröbner bases for polynomial systems with parameters
Journal of Symbolic Computation
Computing comprehensive Gröbner systems and comprehensive Gröbner bases simultaneously
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Proving geometric theorems by partitioned-parametric gröbner bases
ADG'04 Proceedings of the 5th international conference on Automated Deduction in Geometry
Comprehensive triangular decomposition
CASC'07 Proceedings of the 10th international conference on Computer Algebra in Scientific Computing
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A new efficient algorithm for computing a comprehensive Grobnersystem 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 Grobnersystems 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 Grobnerbasis 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 Grobnersystem, 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.