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Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 234–255 | Cite as

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

  • Dong LuEmail author
  • Yao Sun
  • Dingkang Wang
Article

Abstract

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.

Keywords

Comprehensive Gröbner basis comprehensive Gröbner system discovering geometric theorems mechanically parametric polynomial system quantifier elimination 

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References

  1. [1]
    Donald B R, Kapur D, and Mundy J L, Symbolic and numerical computation for artificial intelligence, Computational Mathematics and Applications, Academic Press, Orlando, Florida, 1992, 52–55.Google Scholar
  2. [2]
    Gao X S and Chou S C, Solving parametric algebraic systems, Proceedings of International Symposium on Symbolic and Algebraic Computation, ACM Press, New York, 1992, 335–341.Google Scholar
  3. [3]
    William Y S, An algorithm for solving parametric linear systems, Journal of Symbolic Computation, 1992, 13(4): 353–394.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Chen C, Golubitsky O, Lemaire F, et al., Comprehensive triangular decomposition, International Workshop on Computer Algebra in Scientific Computing, Springer, Berlin, 2007, 73–101.CrossRefGoogle Scholar
  5. [5]
    Lazard D and Rouillier F, Solving parametric polynomial systems, Journal of Symbolic Computation, 2007, 42(6): 636–667.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Huang Z, Parametric equation solving and quantifier elimination in finite fields with the characteristic set method, Journal of Systems Science and Complexity, 2012, 25(4): 778–791.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Chen Z H, Tang X X, and Xia B C, Generic regular decompositions for parametric polynomial systems, Journal of Systems Science and Complexity, 2015, 28(5): 1194–1211.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Chen X F, Li P, Lin L, et al., Proving geometric theorems by partitioned-parametric Gröbner bases, International Workshop on Automated Deduction in Geometry, 2004, 34–43.Google Scholar
  9. [9]
    Lin L, Automated geometric theorem proving and parametric polynomial equations solving, Master Degree Thesis, Institute of Systems Science, CAS, Beijing, 2006.Google Scholar
  10. [10]
    Wang D K and Lin L, Automatic discovering of geometric theorems by computing Gröbner bases with parameters. The 11th Internatinal Conference on Applications of Computer Algebra, 2005.Google Scholar
  11. [11]
    Montes A and Recio T, Automatic discovery of geometry theorems using minimal canonical comprehensive Gröbner systems, International Workshop on Automated Deduction in Geometry, 2006, 113–138.Google Scholar
  12. [12]
    Zhou J, Wang D K, and Sun Y, Automated reducible geometric theorem proving and discovery by Gröbner basis method, Journal of Autotamed Reasoning, 2017, 59(3): 331–344.CrossRefzbMATHGoogle Scholar
  13. [13]
    Botana F, Montes A, and Recio T, An algorithm for automatic discovery of algebraic loci, International Workshop on Automated Deduction in Geometry, 2012, 53–59.Google Scholar
  14. [14]
    Gao X S, Hou X, Tang J, et al., Complete solution classification for the perspective-three-point problem, IEEE Trans. Pattern Anal. Mach. Intell., 2003, 25(8): 930–943.CrossRefGoogle Scholar
  15. [15]
    Zhou J and Wang D K, Solving the perspective-three-point problem using comprehensive Gröbner systems, Journal of Systems Science and Complexity, 2016, 29(5): 1446–1471.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Weispfenning V, A new approach to quantifier elimination for real algebra, Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer, 1998, 376–392.CrossRefGoogle Scholar
  17. [17]
    Kapur D, A quantifier-elimination based heuristic for automatically generating inductive assertions for programs, Journal of Systems Science and Complexity, 2006, 19(3): 307–330.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Fukasaku R, Iwane H, and Sato Y, Real quantifier elimination by computation of comprehensive Gröbner systems, Proceedings of International Symposium on Symbolic and Algebraic Computation, ACM Press, Bath, 2015, 173–180.Google Scholar
  19. [19]
    Fukasaku R, Inoue S, and Sato Y, On QE algorithms over an algebraically closed field based on comprehensive Gröbner systems, Mathematics in Computer Science, 2015, 9(3): 267–281.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Fukasaku R, Iwane H, and Sato Y, Improving a CGS-QE algorithm, Revised Selected Papers of the International Conference on Mathematical Aspects of Computer and Information Sciences, Springer-Verlag, New York, 2015, 231–235.Google Scholar
  21. [21]
    Fukasaku R, Iwane H, and Sato Y, On the implementation of CGS real QE, International Congress on Mathematical Software, Springer International Publishing, 2016, 165–172.Google Scholar
  22. [22]
    Weispfenning V, Comprehensive Gröbner bases, Journal of Symbolic Computation, 1992, 14(1): 1–29.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Pesh M, Computing comprehensive Gröbner bases using MAS, User Manual, 1994.Google Scholar
  24. [24]
    Kapur D, An approach for solving systems of parametric polynomial equations, Principles and Practice of Constraint Programming, MIT Press, Cambridge, Massachusetts, 1995, 217–224.Google Scholar
  25. [25]
    Montes A, A new algorithm for discussing Gröbner bases with parameters, Journal of Symbolic Computation, 2002, 33(2): 183–208.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Weispfenning V, Canonical comprehensive Gröbner bases, Proceedings of International Symposium on Symbolic and Algebraic Computation, ACM Press, New York, 2002, 270–276.Google Scholar
  27. [27]
    Weispfenning V, Canonical comprehensive Gröbner bases, Journal of Symbolic Computation, 2003, 36(3): 669–683.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Manubens M and Montes A, Improving DISPGB algorithm using the discriminant ideal, J. Symbolic. Comput., 2006, 41(11): 1245–1263.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Suzuki A and Sato Y, An alternative approach to comprehensive Gröbner bases, J. Symbolic. Comput., 2003, 36(3–4): 649–667.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Suzuki A and Sato Y, Comprehensive Gröbner bases via ACGB, The 10th Internatinal Conference on Applications of Computer Algebra, 2004, 65–73.Google Scholar
  31. [31]
    Wibmer M, Gröbner bases for families of affine or projective schemes, J. Symbolic. Comput., 2007, 42(8): 803–834.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Manubens M and Montes A, Minimal canonical comprehensive Gröbner system, J. Symbolic. Comput., 2009, 44(5): 463–478.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    Montes A and Wibmer M, Gröbner bases for polynomial systems with parameters, J. Symbolic. Comput., 2010, 45(12): 1391–1425.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Suzuki A and Sato Y, A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases, Proceedings of International Symposium on Symbolic and Algebraic Computation, 2006, 326–331.Google Scholar
  35. [35]
    Kalkbrener M, On the stability of Gröbner bases under specializations, Journal of Symbolic Computation, 1997, 24(1): 51–58.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    Nabeshima K, A speed-up of the algorithm for computing comprehensive Gröbner systems, Proceedings of International Symposium on Symbolic and Algebraic Computation, 2007, 299–306.Google Scholar
  37. [37]
    Kapur D, Sun Y, and Wang D K, A new algorithm for computing comprehensive Gröbner systems, Proceedings of International Symposium on Symbolic and Algebraic Computation, 2010, 29–36.Google Scholar
  38. [38]
    Kapur D, Sun Y, and Wang D K, An efficient algorithm for computing a comprehensive Gröbner system of a parametric polynomial systems, Journal of Symbolic Computation, 2010, 49: 27–44.CrossRefzbMATHGoogle Scholar
  39. [39]
    Kapur D, Sun Y, and Wang D K, Computing comprehensive Gröbner systems and comprehensive Gröbner bases simultaneously, Proceedings of International Symposium on Symbolic and Algebraic Computation, 2011, 193–200.Google Scholar
  40. [40]
    Kapur D, Sun Y, and Wang D K, An efficient method for computing comprehensive Gröbner bases, Journal of Symbolic Computation, 2013, 52: 124–142.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    Kapur D and Yang Y, An algorithm for computing a minimal comprehensive Gröbner basis of a parametric polynomial system, Proceedings of Conference Encuentros de Algebra Comptacionaly Aplicaciones (EACA), Invited Talk, Barcelona, Spain, 2014, 21–25.Google Scholar
  42. [42]
    Kapur D and Yang Y, An algorithm to check whether a basis of a parametric polynomial system is a comprehensive Gröbner basis and the associated completion algorithm, Proceedings of International Symposium on Symbolic and Algebraic Computation, 2015, 243–250.Google Scholar
  43. [43]
    Kapur D, Comprehensive Gröbner basis theory for a parametric polynomial ideal and the associated completion algorithm, Journal of Systems Science and Complexity, 2017, 30(1): 196–233.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    Hashemi A, Darmian M D, and Barkhordar M, Gröbner systems conversion, Mathematics in Computer Science, 2017, 11(1): 61–77.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    Fukuda K, Jensen A, Lauritzen N, et al., The generic Gröbner walk, J. Symb. Comput., 2007, 42(3): 298–312.CrossRefzbMATHGoogle Scholar
  46. [46]
    Hashemi A, Darmian M D, and Barkhordar M, Universal Gröbner basis for parametric polynomial ideals, The International Congress on Mathematical Software, Springer, Cham, 2018, 191–199.Google Scholar
  47. [47]
    Kurata Y, Improving Suzuki-Sato’s CGS algorithm by using stability of Gröbner bases and basic manipulations for efficient implementation, Communications of the Japan Society for Symbolic and Algebraic Computation, 2011, 1: 39–66.Google Scholar
  48. [48]
    Wu W T, On the decision problem and the mechanization of theorem proving in elementary geometry, Sci. Sin., 1978, 21: 159–172.MathSciNetzbMATHGoogle Scholar
  49. [49]
    Wu W T, Basic principles of mechanical theorem proving in elementary geometries, J. Autom. Reason, 1986, 2(3): 221–252.CrossRefzbMATHGoogle Scholar
  50. [50]
    Cox D, Little J, and O’shea D, Ideals, Varieties, and Algorithms, Springer, New York, 1992.CrossRefzbMATHGoogle Scholar
  51. [51]
    Caviness B F and Johnson J R, Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer Science and Business Media, New York, 2012.Google Scholar
  52. [52]
    Collins G E, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Automata Theory and Formal Languages (Second GI Conf., Kaiserslautern), 1975, 134–183.Google Scholar
  53. [53]
    Wang D K, Mechanical proving of a group of space geometric theorem, Master Degree Thesis, Institute of Systems Science, CAS, Beijing, 1990.Google Scholar
  54. [54]
    Wang D K, A mechanical solution to a group of space geometry problem, Proceedings of the International Workshop on Mathematics Mechanization, 1992, 236–243.Google Scholar
  55. [55]
    Deakin M A B, A simple proof of the Beijing theorem, The Mathematical Gazette, 1992, 76(476): 251–254.CrossRefzbMATHGoogle Scholar
  56. [56]
    Nagasaka K, Parametric greatest common divisors using comprehensive Gröbner systems, Proceedings of International Symposium on Symbolic and Algebraic Computation, 2017, 341–348.Google Scholar
  57. [57]
    Kapur D, Lu D, Monagan M, et al., An efficient algorithm for computing parametric multivariate polynomial GCD, Proceedings of International Symposium on Symbolic and Algebraic Computation, 2018, 239–246.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.KLMM, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.SKLOIS, Institute of Information EngineeringChinese Academy of SciencesBeijingChina

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