A Modified Van der Waerden Algorithm to Decompose Algebraic Varieties and Zero-Dimensional Radical Ideals

  • Jia Li
  • Xiao-Shan Gao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5081)


In this paper, we introduce a modified Van der Waerden algorithm to decompose a variety into the union of irreducible varieties. We give an effective representation for irreducible varieties obtained by the algorithm, which allows us to obtain an irredundant decomposition easily. We show that in the zero dimensional case, the polynomial systems for the irreducible varieties obtained in the Van der Waerden algorithm generate prime ideals. As a consequence, we have an algorithm to decompose the radical ideal generated by a finite set of polynomials as the intersection of prime ideals and the degree of the polynomials in the computation is bounded by O(d n ) where d is the degree of the input polynomials and n is the number of variables.


Algebraic variety zero dimensional variety resultant irredundant decomposition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory. In: Bose, N.K. (ed.) Recent Trends in Multidimensional Systems theory, D.Reidel Publ. Comp. (1985)Google Scholar
  2. 2.
    Canny, J.: Generalised characteristics polynomials. Journal of Symbolic Computation 9, 241–250 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chistov, A.: Algorithm of polynomial complexity for factoring polynomials and finding the components of varieties in subexponential time. J. Sov. Math. 4, 1838–1882 (1986)CrossRefGoogle Scholar
  4. 4.
    Chou, S.C.: Mechanical geometry theorem proving. D.Reidel Publishing Company, Dordrecht (1988)zbMATHGoogle Scholar
  5. 5.
    Chou, S.C., Gao, X.S.: Ritt-Wu’s decomposition algorithm and geometry theorem proving. In: Stickel, M.E. (ed.) CADE 1990. LNCS, vol. 449, pp. 207–220. Springer, Heidelberg (1990)Google Scholar
  6. 6.
    Elkadi, M., Mourrain, B.: A new algorithm for the geometric decomposition of a variety. In: Proc. of ISSAC 1999, pp. 9–16. ACM Press, New York (1999)CrossRefGoogle Scholar
  7. 7.
    Gao, X.S., Chou, S.C.: On the dimension for arbitrary ascending chains. Chinese Bull. of Scis. 38, 396–399 (1993)Google Scholar
  8. 8.
    Gao, X.S., Chou, S.C.: On the theory of resolvents and its applications. Systems Science and Mathematical Sciences 12, 17–30 (1999)zbMATHGoogle Scholar
  9. 9.
    Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomial ideals. Journal of Symbolic Computation 6, 149–167 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Laplagne, S.: An algorithm for the computation of the radical of an ideal. In: Proc. of ISSAC 2006, pp. 191–195. ACM Press, New York (2006)CrossRefGoogle Scholar
  11. 11.
    Macaulay, F.S.: The algebraic theory of modular systems. Cambridge University Press, Cambridge (1916)zbMATHGoogle Scholar
  12. 12.
    Sausse, A.: A new approach to primary decompositon. Journal of Symbolic Computation 31, 243–257 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ritt, J.F.: Differential algebra, American Mathematical Society (1950)Google Scholar
  14. 14.
    Szántó, Á.: Computation with polynomial systems, PhD thesis, Cornell University (1999)Google Scholar
  15. 15.
    Van der Waerden, B.L.: Einfürung in die algebraischen geometrie. Springer, Berlin (1973)Google Scholar
  16. 16.
    Van der Waerden, B.L.: Modern algebra II. Frederick Ungar Pub., New York (1953)Google Scholar
  17. 17.
    Wang, D.M.: Irreducible decomposition of algebraic varieties via characteristic sets and Gröbner bases. Computer Aided Geometric Design 9, 471–484 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wang, D.M.: Decomposing algebraic varieties. In: Wang, D., Yang, L., Gao, X.-S. (eds.) ADG 1998. LNCS (LNAI), vol. 1669, pp. 180–206. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  19. 19.
    Wu, W.T.: Basic principles of mechanical theorem-proving in elementary geometries. Journal of System Science and Mathematical Sciences 4, 207–235 (1984); Re-published in Journal of Automated Reasoning 2, 221–252 (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jia Li
    • 1
  • Xiao-Shan Gao
    • 1
  1. 1.Key Laboratory of Mathematics Mechanization Institute of Systems Science, AMSSChinese Academy of SciencesBeijingChina

Personalised recommendations