Advertisement

Some Properties of Triangular Sets and Improvement Upon Algorithm CharSer

  • Yong-Bin Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4120)

Abstract

We present some new properties of triangular sets, which have rather theoretical contribution to understand the structure of the affine varieties of triangular sets. Based on these results and the famous algorithm CharSet, we present two modified versions of the algorithm CharSer that can decompose any nonempty polynomial set into characteristic series. Some examples show that our improvement can efficiently avoid for redundant decompositions, and reduce the branches of the decomposition tree at times.

Keywords

Triangular sets quasi-normal zero characteristic set algorithm CharSer 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comput. 28, 105–124 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Aubry, P., Moreno Maza, M.: Triangular sets for solving polynomial systems: a comparative implementation of four methods. J. Symb. Comput. 28, 125–154 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    Chou, S.-C., Gao, X.-S.: Computations with parametric equations. In: Proceedings ISAAC 1991, pp. 122–127 (1991)Google Scholar
  5. 5.
    Chou, S.-C., Gao, X.-S.: Solving parametric algebraic systems. In: Ibaraki, T., Iwama, K., Yamashita, M., Inagaki, Y., Nishizeki, T. (eds.) ISAAC 1992. LNCS, vol. 650, pp. 335–341. Springer, Heidelberg (1992)Google Scholar
  6. 6.
    Dahan, X., Moreno Maza, M., Schost, É., Wu, W., Xie, Y.: Lifting techniques for triangular decomposition. In: ISSAC 2005, pp. 108–115 (2005)Google Scholar
  7. 7.
    Gallo, G., Mishra, B.: Efficient algorithms and bounds for Wu-Ritt characteristic sets. In: Proceedings MEGA 1990, pp. 119–142 (1990)Google Scholar
  8. 8.
    Gallo, G., Mishra, B.: Wu-Ritt characteristic sets and their complexity. In: Goodman, J.E., Pollack, R., Steiger, W. (eds.) Discrete and Computational Geometry: Papers from the DIMACS Special Year. Dimacs Series in Discrete Mathematics and Theoretical Computer Science, vol. 6, pp. 111–136 (1991)Google Scholar
  9. 9.
    Kalkbrener, M.: A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. J. Symb. Comput. 15, 143–167 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Lazard, D.: A new method for solving algebraic systems of positive dimension. Discrete Appl. Math. 33, 147–160 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Li, Z.-M.: Determinant polynomial sequences. Chinese Sci. Bull. 34, 1595–1599 (1989)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Li, Y.-B., Zhang, J.-Z., Yang, L.: Decomposing polynomial systems into strong regular sets. In: Proceedings ICMS 2002, pp. 361–371 (2002)Google Scholar
  13. 13.
    Li, Y.-B.: Applications of the theory of weakly nondegenerate conditions to zero decomposition for polynomial systems. J. Symb. Comput. 38, 815–832 (2004)CrossRefGoogle Scholar
  14. 14.
    Li, Y.-B.: An alternative algorithm for computing the pseudo-remainder of multivariate polynomials. Applied Mathematics and Computation 173, 484–492 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Wang, D.: Some improvements on Wu’s method for solving systems of algebraic equations. In: Wen-Tsün, W., Min-De, C. (eds.) Proc. of the Int. Workshop on Math. Mechanisation, Beijing, China (1992)Google Scholar
  16. 16.
    Wang, D.: An elimination method for polynomial systems. J. Symb. Comput. 16, 83–114 (1993)CrossRefzbMATHGoogle Scholar
  17. 17.
    Wang, D.: An implementation of the characteristic set method in Maple. In: Pfalzgraf, J., Wang, D. (eds.) Automated Practical Reasoning: Algebraic Approaches, pp. 187–201. Springer, Wien (1995)Google Scholar
  18. 18.
    Wang, D.: Elimination methods. Springer, Wien (2001)zbMATHGoogle Scholar
  19. 19.
    Wang, D.: Elimination Practice. Imperial College Press, London (2003)Google Scholar
  20. 20.
    Wu, W.-T.: On the decision problem and the mechanization of theorem-proving in elementary geometry. Scientia Sinica 21, 159–172 (1978)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Wu, W.-T.: On zeros of algebraic equations–an application of Ritt principle. Kexue Tongbao 31, 1–5 (1986)zbMATHGoogle Scholar
  22. 22.
    Wu, W.-T.: A zero structure theorem for polynomial equations solving. MM Research Preprints 1, 2–12 (1987)Google Scholar
  23. 23.
    Yang, L., Zhang, J.-Z.: Search dependency between algebraic equations: An algorithm applied to automated reasoning. Technical Report ICTP/91/6, International Center For Theoretical Physics, International Atomic Energy Agency, Miramare, Trieste (1991)Google Scholar
  24. 24.
    Yang, L., Zhang, J.-Z., Hou, X.-R.: Non-Linear equation systems and automated theorem proving. Shanghai Sci & Tech Education Publ. House, Shanghai (1996) (in Chinese)Google Scholar
  25. 25.
    Zhang, J.-Z., Yang, L., Hou, X.-R.: A note on Wu Wen-Tsün’s nondegenerate condition. Technical Report ICTP/91/160, International Center For Theoretical Physics, International Atomic Energy Agency, Miramare, Trieste (1991); Also in Chinese Science Bulletin 38(1), 86–87 (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yong-Bin Li
    • 1
    • 2
  1. 1.Chengdu Institute of Computer ApplicationsChinese Academy of SciencesChengduChina
  2. 2.School of Applied MathematicUniversity of Electronic Science and Technology of ChinaChengduChina

Personalised recommendations