A speed-up of the algorithm for computing comprehensive Gröbner systems

We introduce a new algorithm for computing comprehensive Gröbner systems.There exists the Suzuki-Sato algorithm for computing comprehensive Gröbner systems. The Suzuki-Sato algorithm often creates overmuch cells of the parameter space for comprehensive Gröbner systems. Therefore the computation becomes heavy. However, by using inequations ("not equal zero"), we can obtain different cells. In many cases, this number of cells of parameter space is smaller than that of Suzuki-Sato's. Therefore, our new algorithm is more efficient than Suzuki-Sato's one, and outputs a nice comprehensive Gröbner system. Our new algorithm has been implemented in the computer algebra system Risa/Asir We compare the runtime of our implementation with the Suzuki-Sato algorithm and find our algorithm superior in many cases.

[1]  Antonio Montes,et al.  A New Algorithm for Discussing Gröbner Bases with Parameters , 2002, J. Symb. Comput..

[2]  Volker Weispfenning,et al.  Comprehensive Gröbner Bases , 1992, J. Symb. Comput..

[3]  佐藤 洋祐,et al.  特集 Comprehensive Grobner Bases , 2007 .

[4]  Masayuki Noro,et al.  Risa/Asir—a computer algebra system , 1992, ISSAC '92.

[5]  Michael Kalkbrener,et al.  On the Stability of Gröbner Bases Under Specializations , 1997, J. Symb. Comput..

[6]  Antonio Montes,et al.  Improving the DISPGB algorithm using the discriminant ideal , 2006, J. Symb. Comput..

[7]  Volker Weispfenning,et al.  Canonical comprehensive Gröbner bases , 2002, ISSAC '02.

[8]  Thomas Becker,et al.  On Gröbner bases under specialization , 2005, Applicable Algebra in Engineering, Communication and Computing.

[9]  Akira Suzuki,et al.  A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases , 2006, ISSAC '06.

[10]  Patrizia M. Gianni,et al.  Properties of Gröbner bases under specializations , 1987, EUROCAL.

[11]  Katsusuke Nabeshima,et al.  PGB: a package for computing parametric Gröbner and related objects , 2007, ACCA.

[12]  Thomas Sturm,et al.  REDLOG: computer algebra meets computer logic , 1997, SIGS.

[13]  Katsusuke Nabeshima,et al.  Reduced Gröbner Bases in Polynomial Rings over a Polynomial Ring , 2009, Math. Comput. Sci..

[14]  Akira Suzuki,et al.  An alternative approach to comprehensive Gröbner bases , 2002, ISSAC '02.