A Survey on Algorithms for Computing Comprehensive Gröbner Systems and Comprehensive Gröbner Bases

Weispfenning in 1992 introduced the concepts of comprehensive Gröbner system/basis of a parametric polynomial system, and he also presented an algorithm to compute them. Since then, this research field has attracted much attention over the past several decades, and many efficient algorithms have been proposed. Moreover, these algorithms have been applied to many different fields, such as parametric polynomial equations solving, geometric theorem proving and discovering, quantifier elimination, and so on. This survey brings together the works published between 1992 and 2018, and we hope that this survey is valuable for this research area.

[1]  Yao Sun,et al.  An efficient algorithm for computing a comprehensive Gröbner system of a parametric polynomial system , 2013, J. Symb. Comput..

[2]  Peng Li,et al.  Proving Geometric Theorems by Partitioned-Parametric Gröbner Bases , 2004, Automated Deduction in Geometry.

[3]  Michael Wibmer,et al.  Gröbner bases for polynomial systems with parameters , 2010, J. Symb. Comput..

[4]  B. F. Caviness,et al.  Quantifier Elimination and Cylindrical Algebraic Decomposition , 2004, Texts and Monographs in Symbolic Computation.

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

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

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

[8]  B. Donald,et al.  Symbolic and Numerical Computation for Artificial Intelligence , 1997 .

[9]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[10]  William Y. Sit.  An Algorithm for Solving Parametric Linear Systems , 1992, J. Symb. Comput..

[11]  V. Weispfenning.  A New Approach to Quantifier Elimination for Real Algebra , 1998 .

[12]  Yao Sun,et al.  A new algorithm for computing comprehensive Gröbner systems , 2010, ISSAC.

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

[14]  Deepak Kapur,et al.  A Quantifier-Elimination Based Heuristic for Automatically Generating Inductive Assertions for Programs , 2006, J. Syst. Sci. Complex..

[15]  Zhenyu Huang,et al.  Parametric equation solving and quantifier elimination in finite fields with the characteristic set method , 2012, Journal of Systems Science and Complexity.

[16]  Deepak Kapur,et al.  An Approach for Solving Systems of Parametric Polynomial Equations , 1993 .

[17]  Xiao-Shan Gao,et al.  Solving parametric algebraic systems , 1992, ISSAC '92.

[18]  Fabrice Rouillier,et al.  Solving parametric polynomial systems , 2004, J. Symb. Comput..

[19]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[21]  Hidenao Iwane,et al.  Improving a CGS-QE Algorithm , 2015, MACIS.

[22]  Tomás Recio,et al.  Automatic Discovery of Geometry Theorems Using Minimal Canonical Comprehensive Gröbner Systems , 2006, Automated Deduction in Geometry.

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

[24]  Hidenao Iwane,et al.  On the Implementation of CGS Real QE , 2016, ICMS.

[25]  Changbo Chen,et al.  Comprehensive Triangular Decomposition , 2007, CASC.

[26]  W. Wu.  ON THE DECISION PROBLEM AND THE MECHANIZATION OF THEOREM-PROVING IN ELEMENTARY GEOMETRY , 2008 .

[27]  Yao Sun,et al.  Automated Reducible Geometric Theorem Proving and Discovery by Gröbner Basis Method , 2016, Journal of Automated Reasoning.

[28]  Yosuke Sato,et al.  On QE Algorithms over an Algebraically Closed Field Based on Comprehensive Gröbner Systems , 2015, Math. Comput. Sci..

[29]  Yao Sun,et al.  An Efficient Algorithm for Computing Parametric Multivariate Polynomial GCD , 2018, ISSAC.

[30]  Deepak Kapur,et al.  Comprehensive Gröbner basis theory for a parametric polynomial ideal and the associated completion algorithm , 2017, J. Syst. Sci. Complex..

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

[32]  Michael Wibmer,et al.  Gröbner bases for families of affine or projective schemes , 2006, J. Symb. Comput..

[33]  Jie Zhou,et al.  Solving the perspective-three-point problem using comprehensive Gröbner systems , 2016, Journal of Systems Science and Complexity.

[34]  Yiming Yan,et al.  An Algorithm for Computing a Minimal Comprehensive Gröbner\, Basis of a Parametric Polynomial System , 2020, ArXiv.

[35]  Wenjun Wu,et al.  Basic principles of mechanical theorem proving in elementary geometries , 1986, Journal of Automated Reasoning.

[36]  Yao Sun,et al.  An efficient method for computing comprehensive Gröbner bases , 2013, J. Symb. Comput..

[37]  Katsusuke Nabeshima,et al.  A speed-up of the algorithm for computing comprehensive Gröbner systems , 2007, ISSAC '07.

[38]  Jianliang Tang,et al.  Complete Solution Classification for the Perspective-Three-Point Problem , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[39]  Yosuke Kurata,et al.  Improving Suzuki-Sato's CGS Algorithm by Using Stability of Gröbner Bases and Basic Manipulations for Efficient Implementation , 2012 .

[40]  Antonio Montes,et al.  Minimal canonical comprehensive Gröbner systems , 2006, J. Symb. Comput..

[41]  Yao Sun,et al.  Computing comprehensive Gröbner systems and comprehensive Gröbner bases simultaneously , 2011, ISSAC '11.

[42]  Kosaku Nagasaka,et al.  Parametric Greatest Common Divisors using Comprehensive Gröbner Systems , 2017, ISSAC.

[43]  Michael A. B. Deakin.  A Simple Proof of the Beijing Theorem , 1992 .

[44]  Hidenao Iwane,et al.  Real Quantifier Elimination by Computation of Comprehensive Gröbner Systems , 2015, ISSAC.

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

[46]  Bican Xia,et al.  Generic regular decompositions for parametric polynomial systems , 2013, J. Syst. Sci. Complex..

[47]  Amir Hashemi,et al.  Universal Gröbner Basis for Parametric Polynomial Ideals , 2018, ICMS.

[48]  Komei Fukuda,et al.  The generic Gröbner walk , 2007, J. Symb. Comput..

[49]  Yiming Yang,et al.  An Algorithm to Check Whether a Basis of a Parametric Polynomial System is a Comprehensive Gröbner Basis and the Associated Completion Algorithm , 2015, ISSAC.

[50]  Amir Hashemi,et al.  Gröbner Systems Conversion , 2017, Mathematics in Computer Science.