Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 37–46 | Cite as

Characteristic Decomposition: From Regular Sets to Normal Sets

  • Chenqi MouEmail author
  • Dongming Wang


In this paper it is shown how to transform a regular triangular set into a normal triangular set by computing the W-characteristic set of their saturated ideal and an algorithm is proposed for decomposing any polynomial set into finitely many strong characteristic pairs, each of which is formed with the reduced lexicographic Gr¨obner basis and the normal W-characteristic set of a characterizable ideal.


Characteristic decomposition characteristic pair normal set regular set,W-characteristic set 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors wish to thank an anonymous referee who pointed out some unclarity in an early version of the proof of Theorem 4.4 and provided the authors with helpful suggestions for improving the paper.


  1. [1]
    Ritt J, Differential Algebra, American Mathematical Society, New York, 1950.CrossRefzbMATHGoogle Scholar
  2. [2]
    Wu W-T, On zeros of algebraic equations: An application of Ritt principle, Kexue Tongbao, 1986, 31(1): 1–5.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Wu W-T, Mechanical Theorem Proving in Geometries: Basic Principles, Springer-Verlag, Wien, 1994 [Translated from the Chinese by X. Jin and D. Wang].CrossRefGoogle Scholar
  4. [4]
    Wu W-T, Mathematics Mechanization: Mechanical Geometry Theorem-Proving, Mechanical Geometry Problem-Solving, and Polynomial Equations-Solving, Kluwer Academic Publishers Norwell, MA, USA, 2001.zbMATHGoogle Scholar
  5. [5]
    Aubry P, Lazard D, and Moreno Maza M, On the theories of triangular sets, J. Symbolic Comput., 1999, 28(1–2): 105–124.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Bächler T, Gerdt V, Lange-Hegermann M, et al., Algorithmic Thomas decomposition of algebraic and differential systems, J. Symbolic Comput., 2012, 47(10): 1233–1266.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Chen C and Moreno Maza M, Algorithms for computing triangular decompositions of polynomial systems, J. Symbolic Comput., 2012, 47(6): 610–642.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Chou S C and Gao X S, Ritt-Wu’s decomposition algorithm and geometry theorem proving, Proceedings of CADE-10, Ed. by Stickel M, Springer-Verlag, Berlin Heidelberg, 1990, 207–220.Google Scholar
  9. [9]
    Hubert E, Notes on triangular sets and triangulation-decomposition algorithms I: Polynomial systems, Symbolic and Numerical Scientific Computation, Eds. by Winkler F and Langer U, Springer-Verlag, Berlin Heidelberg, 2003, 143–158.Google Scholar
  10. [10]
    Kalkbrener M, A generalized Euclidean algorithm for computing triangular representations of algebraic varieties, J. Symbolic Comput., 1993, 15(2): 143–167.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Lazard D, A new method for solving algebraic systems of positive dimension, Discrete Appl. Math., 1991, 33(1–3): 147–160.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Wang D, An elimination method for polynomial systems, J. Symbolic Comput., 1993, 16(2): 83–114.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Wang D, Decomposing polynomial systems into simple systems, J. Symbolic Comput., 1998, 25(3): 295–314.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Wang D, Computing triangular systems and regular systems, J. Symbolic Comput., 2000, 30(2): 221–236.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Buchberger B, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal, PhD thesis, Universität Innsbruck, Austria, 1965.zbMATHGoogle Scholar
  16. [16]
    Faugère J C, A new efficient algorithm for computing Gröbner bases (F 4), J. Pure Appl. Algebra, 1999, 139(1–3): 61–88.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Faugère J C, A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5), Proceedings of ISSAC 2002, Ed. by Mora T, ACM Press, 2002, 75–83.Google Scholar
  18. [18]
    Faugère J C, Gianni P, Lazard D, et al., Efficient computation of zero-dimensional Gröbner bases by change of ordering, J. Symbolic Comput., 1993, 16(4): 329–344.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Gianni P, Trager B, and Zacharias G, Gröbner bases and primary decomposition of polynomial ideals, J. Symbolic Comput., 1988, 6(2): 149–167.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Kapur D, Sun Y, and Wang D, A new algorithm for computing comprehensive Gröbner systems, Proceedings of ISSAC 2010, Ed. by Watt S, ACM Press, 2010, 29–36.Google Scholar
  21. [21]
    Gao S, Volny F, and Wang M, A new framework for computing Gröbner bases, Math. Comp., 2016, 85(297): 449–465.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Shimoyama T and Yokoyama K, Localization and primary decomposition of polynomial ideals, J. Symbolic Comput., 1996, 22(3): 247–277.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Weispfenning V, Comprehensive Gröbner bases, J. Symbolic Comput., 1992, 14(1): 1–29.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Buchberger B, Gröbner bases: An algorithmic method in polynomial ideal theory, Multidimensional Systems Theory, Ed. by Bose N, Springer, Netherlands, 1985, 184–232.CrossRefGoogle Scholar
  25. [25]
    Dahan X, On lexicographic Gröbner bases of radical ideals in dimension zero: Interpolation and structure, arXiv: 1207.3887, 2012.Google Scholar
  26. [26]
    Lazard D, Solving zero-dimensional algebraic systems, J. Symbolic Comput., 1992, 13(2): 117–131.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Wang D, On the connection between Ritt characteristic sets and Buchberger-Gröbner bases, Math. Comput. Sci., 2016, 10: 479–492.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Gao X S and Chou S C, Solving parametric algebraic systems, Proceedings of ISSAC 1992, Ed. by Wang P, ACM Press, 1992, 335–341.Google Scholar
  29. [29]
    Li B and Wang D, An algorithm for transforming regular chain into normal chain, Computer Mathematics, Ed. by Kapur D, Springer-Verlag, Berlin Heidelberg, 2008, 236–245.CrossRefGoogle Scholar
  30. [30]
    Wang D and Zhang Y, An algorithm for decomposing a polynomial system into normal ascending sets, Sci. China, Ser. A, 2007, 50(10): 1441–1450.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Dong R and Mou C, Decomposing polynomial sets simultaneously into Gröbner bases and normal triangular sets, Proceedings of CASC 2017, Eds. by Gerdt V, Koepf W, Seiler W, et al., Springer-Verlag, Berlin Heidelberg, 2017, 77–92.Google Scholar
  32. [32]
    Wang D, Dong R, and Mou C, Decomposition of polynomial sets into characteristic pairs, arXiv: 1702.08664, 2017.Google Scholar
  33. [33]
    Mou C, Wang D, and Li X, Decomposing polynomial sets into simple sets over finite fields: The positive-dimensional case, Theoret. Comput. Sci., 2013, 468: 102–113.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Wang D, Elimination Methods, Springer-Verlag, Wien, 2001.CrossRefzbMATHGoogle 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.LMIB – School of Mathematics and Systems Science/Beijing Advanced Innovation Center for Big Data and Brain ComputingBeihang UniversityBeijingChina
  2. 2.LMIB – SKLSDE – School of Mathematics and Systems Science/Beijing Advanced Innovation Center for Big Data and Brain ComputingBeihang UniversityBeijingChina
  3. 3.Centre National de la Recherche ScientifiqueParis cedex 16France

Personalised recommendations