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

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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.

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References

  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. 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. William Y S, An algorithm for solving parametric linear systems, Journal of Symbolic Computation, 1992, 13(4): 353–394.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  5. Lazard D and Rouillier F, Solving parametric polynomial systems, Journal of Symbolic Computation, 2007, 42(6): 636–667.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. Lin L, Automated geometric theorem proving and parametric polynomial equations solving, Master Degree Thesis, Institute of Systems Science, CAS, Beijing, 2006.

    Google Scholar 

  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. 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. 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.

    Article  MATH  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  16. Weispfenning V, A new approach to quantifier elimination for real algebra, Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer, 1998, 376–392.

    Chapter  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Weispfenning V, Comprehensive Gröbner bases, Journal of Symbolic Computation, 1992, 14(1): 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  23. Pesh M, Computing comprehensive Gröbner bases using MAS, User Manual, 1994.

    Google Scholar 

  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. Montes A, A new algorithm for discussing Gröbner bases with parameters, Journal of Symbolic Computation, 2002, 33(2): 183–208.

    Article  MathSciNet  MATH  Google Scholar 

  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. Weispfenning V, Canonical comprehensive Gröbner bases, Journal of Symbolic Computation, 2003, 36(3): 669–683.

    Article  MathSciNet  MATH  Google Scholar 

  28. Manubens M and Montes A, Improving DISPGB algorithm using the discriminant ideal, J. Symbolic. Comput., 2006, 41(11): 1245–1263.

    Article  MathSciNet  MATH  Google Scholar 

  29. Suzuki A and Sato Y, An alternative approach to comprehensive Gröbner bases, J. Symbolic. Comput., 2003, 36(3–4): 649–667.

    Article  MathSciNet  MATH  Google Scholar 

  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. Wibmer M, Gröbner bases for families of affine or projective schemes, J. Symbolic. Comput., 2007, 42(8): 803–834.

    Article  MathSciNet  MATH  Google Scholar 

  32. Manubens M and Montes A, Minimal canonical comprehensive Gröbner system, J. Symbolic. Comput., 2009, 44(5): 463–478.

    Article  MathSciNet  MATH  Google Scholar 

  33. Montes A and Wibmer M, Gröbner bases for polynomial systems with parameters, J. Symbolic. Comput., 2010, 45(12): 1391–1425.

    Article  MathSciNet  MATH  Google Scholar 

  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. Kalkbrener M, On the stability of Gröbner bases under specializations, Journal of Symbolic Computation, 1997, 24(1): 51–58.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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.

    Article  MATH  Google Scholar 

  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. Kapur D, Sun Y, and Wang D K, An efficient method for computing comprehensive Gröbner bases, Journal of Symbolic Computation, 2013, 52: 124–142.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  44. Hashemi A, Darmian M D, and Barkhordar M, Gröbner systems conversion, Mathematics in Computer Science, 2017, 11(1): 61–77.

    Article  MathSciNet  MATH  Google Scholar 

  45. Fukuda K, Jensen A, Lauritzen N, et al., The generic Gröbner walk, J. Symb. Comput., 2007, 42(3): 298–312.

    Article  MATH  Google Scholar 

  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. 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. Wu W T, On the decision problem and the mechanization of theorem proving in elementary geometry, Sci. Sin., 1978, 21: 159–172.

    MathSciNet  MATH  Google Scholar 

  49. Wu W T, Basic principles of mechanical theorem proving in elementary geometries, J. Autom. Reason, 1986, 2(3): 221–252.

    Article  MATH  Google Scholar 

  50. Cox D, Little J, and O’shea D, Ideals, Varieties, and Algorithms, Springer, New York, 1992.

    Book  MATH  Google Scholar 

  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. 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. 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. 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. Deakin M A B, A simple proof of the Beijing theorem, The Mathematical Gazette, 1992, 76(476): 251–254.

    Article  MATH  Google Scholar 

  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. 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 

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Author information

Authors and Affiliations

  1. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Dong Lu & Dingkang Wang

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Dong Lu & Dingkang Wang

  3. SKLOIS, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Yao Sun

Authors
  1. Dong Lu
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  2. Yao Sun
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  3. Dingkang Wang
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Correspondence to Dong Lu.

Additional information

This research was supported in part by the CAS Project QYZDJ-SSW-SYS022, the National Natural Science Foundation of China under Grant No. 61877058, and the Strategy Cooperation Project AQ-1701.

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Lu, D., Sun, Y. & Wang, D. A Survey on Algorithms for Computing Comprehensive Gröbner Systems and Comprehensive Gröbner Bases. J Syst Sci Complex 32, 234–255 (2019). https://doi.org/10.1007/s11424-019-8357-z

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  • DOI: https://doi.org/10.1007/s11424-019-8357-z

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