Science in China Series A: Mathematics

, Volume 50, Issue 10, pp 1441–1450 | Cite as

An algorithm for decomposing a polynomial system into normal ascending sets

  • Ding-kang Wang
  • Yan Zhang


We present an algorithm to decompose a polynomial system into a finite set of normal ascending sets such that the set of the zeros of the polynomial system is the union of the sets of the regular zeros of the normal ascending sets. If the polynomial system is zero dimensional, the set of the zeros of the polynomials is the union of the sets of the zeros of the normal ascending sets.


zero decomposition normal ascending set polynomial system 


68Q40 13P10 


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  1. [1]
    Ritt J F. Differential Algebra. New York: American Mathematical Society, 1950MATHGoogle Scholar
  2. [2]
    Wu W T. Basic principles of mechanical theorem proving in elementary geometries. J Sys Sci & Math Sci, 4: 20–235 (1984)Google Scholar
  3. [3]
    Chou S C, Mechanical Geometry Theorem-proving, Dordrecht: D. Reidel Pub. Company, 1988MATHGoogle Scholar
  4. [4]
    Gao X S, Chou S C. The dimension of ascending chains. Chin Sci Bull, 38(5): 396–399 (1993)Google Scholar
  5. [5]
    Gao X S, Chou S C. Ritt-Wu’s decomposition algorithm and geometry theorem proving. In: Proceedings of CADE-10, Lecture Notes in Artificial Intelligence. 449: 207–220 (1990)Google Scholar
  6. [6]
    Wang D M. Elimination Method. Wien-New York: Springer-Verlag, (2001)Google Scholar
  7. [7]
    Gao X S, Chou S C. Solving parametric algebraic systems. In: Proceedings of ISSAC’92, 1992, 335–341Google Scholar
  8. [8]
    Trager B M. Algebraic factoring and rational function integration. In: Proceedings of ACM SYMSAC, 1976, 219–226Google Scholar
  9. [9]
    Yuan C M. Trager’s factorization algorithm over successive extension field. J Sys Sci & Math Scis, 26(5): 53–40 (2006)Google Scholar
  10. [10]
    Yang L, Zhang J Z. Searching dependency between algebraic equations: An algorithm applied to automated reasoning. In: Johnson J, McKee S, Vella A, eds. Artificial Intelligence in Mathematics. Oxford: Oxford University Press, 1994, 147–156Google Scholar
  11. [11]
    Kalbrener M. A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. J Sym Comput, 15: 143–167 (1993)CrossRefGoogle Scholar
  12. [12]
    Lazard D. A new method for solving algebraic systems of positive dimension. Discrete Appl Math, 33: 147–160 (1991)MATHCrossRefGoogle Scholar
  13. [13]
    Maza M M. On triangular decompositions of algebraic varieties. Technical Report TR 4/99, NAG Ltd, Oxford, UK, 1999Google Scholar
  14. [14]
    Aubry P, Lazard D, Maza M M. On the theories of triangular sets. J Sym Comput, 28: 105–124 (1999)MATHCrossRefGoogle Scholar
  15. [15]
    Szanto A. Computation with polynomial systems. Dissertation for the Doctoral Degree. Cornell: Cornell University, 1999Google Scholar
  16. [16]
    Kandri R A, Maarouf H, Ssafini M. Triviality and dimension of a system of algebraic differential equations. J Aut Rea, 20: 365–385 (1998)MATHCrossRefGoogle Scholar
  17. [17]
    Bouziane D, Kandri R A, Maarouf H. Unmixed-dimensional decomposition of a finitely generated perfect differential ideal. J Sym Comput, 31: 631–649 (2001)MATHCrossRefGoogle Scholar
  18. [18]
    Loos R. Computing in algebraic extensions. In: Buchberger B, Collins G E, Loos R, eds. Computer Algebra: Symbolic and Algebraic Computation. 2nd ed. Wien-New York: Springer-Verlag, 1983, 173–188Google Scholar
  19. [19]
    Cox D, Little J, Shea D O’. Ideals, Varieties, and Algorithms. 2nd ed. New York: Springer-Verlag, 1997, 149–159Google Scholar

Copyright information

© Science in China Press 2007

Authors and Affiliations

  1. 1.Key Laboratory of Mathematics MechanizationAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina

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