Skip to main content
SpringerLink
Log in

On QE Algorithms over an Algebraically Closed Field Based on Comprehensive Gröbner Systems

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

We introduce new algorithms for quantifier eliminations (QE) in the domain of an algebraically closed field. Our algorithms are based on the computation of comprehensive Gröbner systems (CGS). We study Suzuki–Sato’s CGS computation algorithm and its successors in more detail and modify them into an optimal form for applying to QE. Based on this modified algorithm, we introduce two QE algorithms. One is pursuing the simplest output quantifier free formula which employs only the computations of CGS. The other consists of parallel computations of CGS and GCD of parametric unary polynomials. It achieves faster computation time with reasonably simple output quantifier free formulas. Our implementation shows that in many examples our algorithms are superior to other existing algorithms such as a QE algorithm adopted in the Mathematica package Reduce and Resolve or a QE algorithm adopted in the Maple package Projection which are the most efficient existing implementations as far as we know.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A computer algebra system Risa/Asir. http://www.math.kobe-u.ac.jp/Asir/asir.html

  2. Basu, S., Pollack, R., Roy, M.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin

  3. Chen, C., Golubitsky, O., Lemaire, F., Moreno Maza, M., Pan, W.: Comprehensive Triangular Decomposition, vol. 4770 of LNCS, pp. 73–101. Springer, Berlin (2007)

  4. Chou S.C.: Mechanical Geometry Theorem Proving. Mathematics and Its Applications. Springer, Berlin (1988)

    Google Scholar 

  5. Fortuna E., Gianni P., Trager B.: Degree reduction under specialization. J. Pure Appl. Algebra 164, 153–164 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gao, X., Wang, D.: Zero decomposition theorems for counting the number of solutions for parametric equation systems. In: Proceedings of ASCM 2003, pp. 129–144 (2003)

  7. Harrison, J.: Complex Quantifier Elimination in HOL.(2001). In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs Supplemental Proceedings, pp. 159–174. Division of Informatics, University of Edinburgh, 2001. Published as Informatics Report Series EDI-INF-RR-0046. http://www.informatics.ed.ac.uk/publications/report/0046.html

  8. A computer algebra system HOL Light. http://www.cl.cam.ac.uk/~jrh13/hol-light/

  9. International Mathematical Olympiad. http://www.imo-official.org/problems.aspx

  10. Kalkbrener M.: On the stability of Gröbner bases under specializations. J. Symb. Comput. 24/1, 51–58 (1997)

    Article  MathSciNet  Google Scholar 

  11. Kapur, D., Sun, Y., Wang, D.: A new algorithm for computing comprehensive Gröbner systems. In: International Symposium on Symbolic and Algebraic Computation, pp. 29–36. ACM-Press (2010)

  12. Kurata Y.: Improving Suzuki–Sato’s CGS algorithm by using stability of Gröbner bases and basic manipulations for efficient implementation. Commun. JSSAC 1, 39–66 (2011)

    Google Scholar 

  13. Mathematica Tutorial: tutorial/ComplexPolynomialSystems

  14. Nabeshima, K.: A speed-up of the algorithm for computing comprehensive Gröbner systems. In: International Symposium on Symbolic and Algebraic Computation, pp. 299–306. ACM-Press (2007)

  15. Nabeshima K.: Stability conditions of monomial bases and comprehensive Gröbner systems. Lect. Notes Comput. Sci. 7442, 248–259 (2012)

    Article  Google Scholar 

  16. Suzuki, A., Sato, Y.: A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases. In: International Symposium on Symbolic and Algebraic Computation, pp. 326–331. ACM-Press (2006)

  17. Wang, D.: The projection property of regular systems and its application to solving parametric polynomial systems. In: Dolzmann, A., Seidl, A., Sturm, T. Algorithmic Algebra and Logic, pp. 269–274. Herstellung und Verlag, Norderstedt (2004)

  18. Weispfenning, V.: Comprehensive Göbner Bases. J. Symb. Comput. 14(1), 1–29 (1992)

Download references

Author information

Authors and Affiliations

  1. Graduate School of Science, Tokyo University of Science, Tokyo, Japan

    Ryoya Fukasaku

  2. Tokyo University of Science, Tokyo, Japan

    Shutaro Inoue & Yosuke Sato

Authors
  1. Ryoya Fukasaku
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shutaro Inoue
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yosuke Sato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yosuke Sato.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fukasaku, R., Inoue, S. & Sato, Y. On QE Algorithms over an Algebraically Closed Field Based on Comprehensive Gröbner Systems. Math.Comput.Sci. 9, 267–281 (2015). https://doi.org/10.1007/s11786-015-0237-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-015-0237-x

Keywords

Mathematics Subject Classification

Search

Navigation