Skip to main content
SpringerLink
Log in

On the Implementation of CGS Real QE

  • Conference paper
  • First Online:
Mathematical Software – ICMS 2016 (ICMS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9725))

Included in the following conference series:

Abstract

A CGS real QE method is a real quantifier elimination (QE) method which is composed of the computation of comprehensive Gröbner systems (CGSs) based on the theory of real root counting. Its fundamental algorithm was first introduced by Weispfenning in 1998. We further improved the algorithm in 2015 so that we can make a satisfactorily practical implementation. For its efficient implementation, there are several key issues we have to take into account. In this extended abstract we introduce them together with some important techniques for making an efficient CGS real QE implementation.

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 EPUB and 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

References

  1. Becker, E., Wörmann, T.: On the trace formula for quadratic forms. In: Recent Advances in Real Algebraic Geometry and Quadratic Forms, Berkeley, CA, 1990/1991; San Francisco, CA, 1991, pp. 271–291. Contemporary Mathematics, vol. 155, American Mathematical Society Providence, RI (1994)

    Google Scholar 

  2. The first prototype CGS real QE program released (2015). http://www.mi.kagu.tus.ac.jp/~fukasaku/issac2015

  3. The latest prototype CGS real QE program released (2016). http://www.mi.kagu.tus.ac.jp/~fukasaku/CGSQE2016

  4. Chen, C., Maza, M.M.: Quantifier elimination by cylindrical algebraic decomposition based on regular chains. In: Proceedings of International Symposium on Symbolic and Algebraic Computation (ISSAC 2014), pp. 91–98. ACM (2014)

    Google Scholar 

  5. Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) Automata Theory and Formal Languages. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)

    Google Scholar 

  6. Dolzmann, A., Gilch, L.A.: Generic hermitian quantifier elimination. In: Buchberger, B., Campbell, J. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 80–93. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  7. Fukasaku, R., Iwane, H., Sato, Y.: Real quantifier elimination by computation of comprehensive Gröbner systems. In: Proceedings of International Symposium on Symbolic and Algebraic Computation (ISSAC 2015), pp. 173–180. ACM (2015)

    Google Scholar 

  8. Fukasaku, R., Iwane, H., Sato, Y.: Improving a CGS-QE algorithm. In: Kotsireas, I.S., Rump, S.M., Yap, C.K. (eds.) MACIS 2015. LNCS, vol. 9582, pp. 231–235. Springer, Heidelberg (2016). doi:10.1007/978-3-319-32859-1_20

    Chapter  Google Scholar 

  9. Mathematica Tutorial (RealPolynomialSystems). http://reference.wolfram.com/language/tutorial/RealPolynomialSystems.html

  10. Pedersen, P., Roy, M.-F., Szpirglas, A.: Counting real zeroes in the multivariate case. In: Proceedings of the Effective Methods in Algebraic Geometry, pp. 203–224 (1993)

    Google Scholar 

  11. QEPCAD-Quantifier elimination by partial cylindrical algebraic decomposition. http://www.usna.edu/CS/qepcadweb/B/QEPCAD.html

  12. Redlog: an integral part of the interactive computer algebra system reduce. http://www.redlog.eu/

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

    Google Scholar 

  14. SyNRAC: a software package for quantifier elimination. http://www.fujitsu.com/jp/group/labs/en/resources/tech/freeware/synrac/

  15. Weispfenning, V.: A new approach to quantifier elimination for real algebra. In: Caviness, B.F., Johnson, J.R. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 376–392. Springer, Vienna (1998)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tokyo University of Science, Tokyo, Japan

    Ryoya Fukasaku

  2. Fujitsu Laboratories LTD/National Institute of Informatics, Tokyo, Japan

    Hidenao Iwane

  3. Tokyo University of Science, Tokyo, Japan

    Yosuke Sato

Authors
  1. Ryoya Fukasaku
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hidenao Iwane
    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 Ryoya Fukasaku .

Editor information

Editors and Affiliations

  1. Universität Kaiserslautern , Kaiserslautern, Germany

    Gert-Martin Greuel

  2. Zuse Institute Berlin , Berlin, Germany

    Thorsten Koch

  3. Johannes Kepler University Linz , Hagenberg, Austria

    Peter Paule

  4. University of Notre Dame , Notre Dame, Indiana, USA

    Andrew Sommese

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Fukasaku, R., Iwane, H., Sato, Y. (2016). On the Implementation of CGS Real QE. In: Greuel, GM., Koch, T., Paule, P., Sommese, A. (eds) Mathematical Software – ICMS 2016. ICMS 2016. Lecture Notes in Computer Science(), vol 9725. Springer, Cham. https://doi.org/10.1007/978-3-319-42432-3_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-42432-3_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-42431-6

  • Online ISBN: 978-3-319-42432-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation