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
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
A signature-based algorithm for computing Gröbner bases in solvable polynomial algebras
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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A new approach is proposed for computing a comprehensive Grobner basis of a parameterized polynomial system. The key new idea is not to simplify a polynomial under various specialization of its parameters, but rather keep track in the polynomial, of the power products whose coefficients vanish; this is achieved by partitioning the polynomial into two parts-nonzero part and zero part for the specialization under consideration. During the computation of a comprehensive Grobner system, for a particular branch corresponding to a specialization of parameter values, nonzero parts of the polynomials dictate the computation, i.e., computing S-polynomials as well as for simplifying a polynomial with respect to other polynomials; but the manipulations on the whole polynomials (including their zero parts) are also performed. Once a comprehensive Grobner system is generated, both nonzero and zero parts of the polynomials are collected from every branch and the result is a faithful comprehensive Grobner basis, to mean that every polynomial in a comprehensive Grobner basis belongs to the ideal of the original parameterized polynomial system. This technique should be applicable to all algorithms for computing a comprehensive Grobner system, thus producing both a comprehensive Grobner system as well as a faithful comprehensive Grobner basis of a parameterized polynomial system simultaneously. To propose specific algorithms for computing comprehensive Grobner bases, a more generalized theorem is presented to give a more generalized stable condition for parametric polynomial systems. Combined with the new approach, the new theorem leads to two efficient algorithms for computing comprehensive Grobner bases. The timings on a collection of examples demonstrate that both these two new algorithms for computing comprehensive Grobner bases have better performance than other existing algorithms.