Skip to main content
SpringerLink
Log in

Some Properties of Triangular Sets and Improvement Upon Algorithm CharSer

  • Conference paper
Book cover Artificial Intelligence and Symbolic Computation (AISC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNAI,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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comput. 28, 105–124 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Chou, S.-C., Gao, X.-S.: Computations with parametric equations. In: Proceedings ISAAC 1991, pp. 122–127 (1991)

    Google Scholar 

  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. 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. Gallo, G., Mishra, B.: Efficient algorithms and bounds for Wu-Ritt characteristic sets. In: Proceedings MEGA 1990, pp. 119–142 (1990)

    Google Scholar 

  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. Kalkbrener, M.: A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. J. Symb. Comput. 15, 143–167 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lazard, D.: A new method for solving algebraic systems of positive dimension. Discrete Appl. Math. 33, 147–160 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, Z.-M.: Determinant polynomial sequences. Chinese Sci. Bull. 34, 1595–1599 (1989)

    MathSciNet  MATH  Google Scholar 

  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. Li, Y.-B.: Applications of the theory of weakly nondegenerate conditions to zero decomposition for polynomial systems. J. Symb. Comput. 38, 815–832 (2004)

    Article  Google Scholar 

  14. Li, Y.-B.: An alternative algorithm for computing the pseudo-remainder of multivariate polynomials. Applied Mathematics and Computation 173, 484–492 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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. Wang, D.: An elimination method for polynomial systems. J. Symb. Comput. 16, 83–114 (1993)

    Article  MATH  Google Scholar 

  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. Wang, D.: Elimination methods. Springer, Wien (2001)

    MATH  Google Scholar 

  19. Wang, D.: Elimination Practice. Imperial College Press, London (2003)

    Google Scholar 

  20. Wu, W.-T.: On the decision problem and the mechanization of theorem-proving in elementary geometry. Scientia Sinica 21, 159–172 (1978)

    MathSciNet  MATH  Google Scholar 

  21. Wu, W.-T.: On zeros of algebraic equations–an application of Ritt principle. Kexue Tongbao 31, 1–5 (1986)

    MATH  Google Scholar 

  22. Wu, W.-T.: A zero structure theorem for polynomial equations solving. MM Research Preprints 1, 2–12 (1987)

    Google Scholar 

  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. 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. 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 

Download references

Author information

Authors and Affiliations

  1. Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu, Sichuan, 610041, China

    Yong-Bin Li

  2. School of Applied Mathematic, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610054, China

    Yong-Bin Li

Authors
  1. Yong-Bin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, 76128, Karlsruhe, Germany

    Jacques Calmet

  2. Department of Computer Science, University of Tsukuba, 305-8573, Tsukuba, Japan

    Tetsuo Ida

  3. LMIB – SKLSDE – School of Science, Beihang University, 100083, Beijing, China

    Dongming Wang

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Li, YB. (2006). Some Properties of Triangular Sets and Improvement Upon Algorithm CharSer. In: Calmet, J., Ida, T., Wang, D. (eds) Artificial Intelligence and Symbolic Computation. AISC 2006. Lecture Notes in Computer Science(), vol 4120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11856290_9

Download citation

  • DOI: https://doi.org/10.1007/11856290_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-39728-1

  • Online ISBN: 978-3-540-39730-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation