Skip to main content
SpringerLink
Log in

Quantifier Elimination for Quartics

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

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4120))

Abstract

Concerning quartics, two particular quantifier elimination (QE) problems of historical interests and practical values are studied. We solve the problems by the theory of complete discrimination systems and negative root discriminant sequences for polynomials that provide a method for real (positive/negative) and complex root classification for polynomials. The equivalent quantifier-free formulas are obtained mainly be hand and are simpler than those obtained automatically by previous methods or QE tools. Also, applications of the results to program verification and determination of positivity of symmetric polynomials are showed.

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. Anai, H., Hara, S.: Fixed-structure robust controller synthesis based on sign definite condition by a special quantifier elimination. In: Proc. American Control Conference 2000, pp. 1312–1316 (2000)

    Google Scholar 

  2. Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition I: The basic algorithm. SIAM J. Comput. 13, 865–877 (1984)

    Article  MathSciNet  Google Scholar 

  3. Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition II: An adjacency algorithm for the plane. SIAM J. Comput. 13, 878–889 (1984)

    Article  MathSciNet  Google Scholar 

  4. Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition III: An adjacency algorithm for three-dimensional space. J. Symb. Comput., 163–187 (1988)

    Google Scholar 

  5. Arnon, D.S., Mignotte, M.: On mechanical quantifier elimination for elementary algebra and geometry. J. Symb. Comput. 5, 237–260 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Besson, F., Jensen, T., Talpin, J.-P.: Polyhedral analysis of synchronous languages. In: Cortesi, A., Filé, G. (eds.) SAS 1999. LNCS, vol. 1694, pp. 51–69. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  7. Brown, C.W.: Simple CAD construction and its applications. J. Symb. Comput. 31, 521–547 (2001)

    Article  MATH  Google Scholar 

  8. Brown, C.W.: Improved projection for cylindrical algebraic decomposition. J. Symb. Comput. 32, 447–465 (2001)

    Article  MATH  Google Scholar 

  9. Brown, C.W., McCallum, S.: On Using Bi-equational Constraints in CAD Construction. In: Kauers, M. (ed.) Proc. ISSAC 2005, pp. 76–83. ACM Press, New York (2005)

    Chapter  Google Scholar 

  10. Caviness, B.F., Johnson, J.R. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Heidelberg (1998)

    MATH  Google Scholar 

  11. Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–165. Springer, Heidelberg (1975)

    Google Scholar 

  12. Collins, G.E.: Quantifier elimination by cylindrical algebraic decomposition - 20 years of progress. In: Caviness, B., Johnson, J. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 8–23. Springer, New York (1998)

    Google Scholar 

  13. Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12, 299–328 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cousot, P.: Abstract interpretation based formal methods and future challenges. In: Wilhelm, R. (ed.) Informatics: 10 Years Back, 10 Years Ahead. LNCS, vol. 2000, pp. 138–156. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  15. Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among the variables of a program. In: ACM POPL 1978, pp. 84–97 (1978)

    Google Scholar 

  16. Dolzmann, A., Sturm, T., Weispfenning, V.: Real quantifier elimination in practice. In: Matzat, B.H., Greuel, G.-M., Hiss, G. (eds.) Algorithmic Algebra and Number Theory, pp. 221–247. Springer, Heidelberg (1998)

    Google Scholar 

  17. Gantmacher, F.R.: The Theory of Matrices. Chelsea Publishing Company, New York (1959)

    MATH  Google Scholar 

  18. González-Vega, L.: A Combinatorial Algorithm Solving Some Quantifier Elimination Problems. In: Caviness, B., Johnson, J. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 365–375. Springer, New York (1998)

    Google Scholar 

  19. González-Vega, L., Lombardi, H., Recio, T., Roy, M.-F.: Sturm-Habicht sequence. In: Proc. of ISSAC 1989, pp. 136–146. ACM Press, New York (1989)

    Chapter  Google Scholar 

  20. Halbwachs, N., Proy, Y.E., Roumanoff, P.: Verification of real-time systems using linear relation analysis. Formal Methods in System Design 11(2), 157–185 (1997)

    Article  Google Scholar 

  21. Henzinger, T.A., Ho, P.-H.: Algorithmic analysis of nonlinear hybrid systems. In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, pp. 225–238. Springer, Heidelberg (1995)

    Google Scholar 

  22. Hong, H.: An improvement of the projection operator in cylindrical algebraic decomposition. In: Watanabe, S., Nagata, M. (eds.) Proceedings of ISSAC 1990, pp. 261–264. ACM Press, New York (1990)

    Chapter  Google Scholar 

  23. Hong, H.: Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination. In: Wang, P.S. (ed.) Proceedings of ISSAC 1992, pp. 177–188. ACM Press, New York (1992)

    Chapter  Google Scholar 

  24. Hong, H.: An efficient method for analyzing the topology of plane real algebraic curves. Math. Comput. Simul. 42, 571–582 (1996)

    Article  MATH  Google Scholar 

  25. Lazard, D.: Quantifier elimination: optimal solution for two classical examples. J. Symb. Comput. 5, 261–266 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  26. McCallum, S.: An improved projection operation for cylindrical algebraic decomposition of three-dimensional space. J. Symb. Comput., 141–161 (1988)

    Google Scholar 

  27. McCallum, S.: An improved projection operator for cylindrical algebraic decomposition. In: Caviness, B., Johnson, J. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 242–268. Springer, New York (1998)

    Google Scholar 

  28. Renegar, J.: On the Computational Complexity and Geometry of the First-order Theory of the Reals. Parts I, II, and III. J. Symb. Comput. 13, 255–352 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tarski, A.: A Decision for Elementary Algebra and Geometry. University of California Press, Berkeley (1951)

    MATH  Google Scholar 

  30. Timofte, V.: On the positivity of symmetric polynomial functions. Part I: General results. J. Math. Anal. Appl. 284, 174–190 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tiwari, A.: Termination of linear programs. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 70–82. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  32. Wang, L., Yu, W.: Complete characterization of strictly positive real regions and robust strictly positive real synthesis method. Science in China 43(E), 97–112 (2000)

    MATH  Google Scholar 

  33. Wang, Z.H., Hu, H.Y.: Delay-independent stability of retarded dynamic systems of multiple degrees of freedom. Journal of Sound and Vibration 226(1), 57–81 (1999)

    Article  MathSciNet  Google Scholar 

  34. Wang, Z.H., Hu, H.Y.: Stability of time-delayed dynamic systems with unknown parameters. Journal of Sound and Vibration 233(2), 215–233 (2000)

    Article  MathSciNet  Google Scholar 

  35. Weispfenning, V.: Quantifier elimination for real algebra - the cubic case. In: Proc. ISSAC 1994, pp. 258–263. ACM Press, Oxford (1994)

    Chapter  Google Scholar 

  36. Weispfenning, V.: Quantifier elimination for real algebra - the quadratic case and beyond. In: AAECC, vol. 8, pp. 85–101 (1997)

    Google Scholar 

  37. Weispfenning, V.: A New Approach to Quantifier Elimination for Real Algebra. In: Caviness, B., Johnson, J. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 376–392. Springer, New York (1998)

    Google Scholar 

  38. Wu, W.T.: On problems involving inequalities. In: MM-Preprints, (7), pp. 1–13 (1992)

    Google Scholar 

  39. Yang, L.: Recent advances on determining the number of real roots of parametric polynomials. J. Symbolic Computation 28, 225–242 (1999)

    Article  MATH  Google Scholar 

  40. Yang, L., Hou, X., Xia, B.: A complete algorithm for automated discovering of a class of inequality-type theorems. Sci. in China (Ser. F) 44, 33–49 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yang, L., Hou, X., Zeng, Z.: A complete discrimination system for polynomials. Science in China (Ser. E) 39, 628–646 (1996)

    MathSciNet  MATH  Google Scholar 

  42. Yang, L., Xia, B.: An explicit criterion to determine the number of roots of a polynomial on an interval. Progress in Natural Science 10(12), 897–910 (2000)

    MathSciNet  Google Scholar 

  43. Yang, L., Xia, B.: Real solution classifications of a class of parametric semi-algebraic systems. In: Proc. of Int’l. Conf. on Algorithmic Algebra and Logic, pp. 281–289 (2005)

    Google Scholar 

  44. Yang, L., Zhan, N., Xia, B., Zhou, C.: Program Verification by Using Discoverer. In: Proc. of the IFIP Working Conference on Verified Software: Tools, Techniques and Experiments, Zurich, October 10-13 (to appear, 2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Guangzhou University, Guangzhou, 510006, China

    Lu Yang

  2. School of Mathematical Sciences, Peking University, China

    Bican Xia

Authors
  1. Lu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bican Xia
    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

Yang, L., Xia, B. (2006). Quantifier Elimination for Quartics. 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_13

Download citation

  • DOI: https://doi.org/10.1007/11856290_13

  • 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