Skip to main content

Generic Hermitian Quantifier Elimination

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 3249))

Abstract

We present a new method for generic quantifier elimination that uses an extension of Hermitian quantifier elimination. By means of sample computations we show that this generic Hermitian quantifier elimination is, for instance, an important method for automated theorem proving in geometry.

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

Buying options

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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Collins, G.E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)

    Google Scholar 

  2. Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation 12, 299–328 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brown, C.W.: Simplification of truth-invariant cylindrical algebraic decompositions. In: Gloor, O. (ed.) Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation (ISSAC 1998), Rostock, Germany, pp. 295–301. ACM Press, New York (1998)

    Chapter  Google Scholar 

  4. Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5, 3–27 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  5. Loos, R., Weispfenning, V.: Applying linear quantifier elimination. THE Computer Journal 36, 450–462 (1993); Special issue on computational quantifier elimination

    Article  MATH  MathSciNet  Google Scholar 

  6. Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31, 2–9 (1997)

    Article  Google Scholar 

  7. Weispfenning, V.: A new approach to quantifier elimination for real algebra. In: Caviness, B., Johnson, J. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation, pp. 376–392. Springer, Wien (1998)

    Google Scholar 

  8. Dolzmann, A.: Reelle Quantorenelimination durch parametrisches Zählen von Nullstellen. Diploma thesis, Universität Passau, D-94030 Passau, Germany (1994)

    Google Scholar 

  9. Gilch, L.A.: Effiziente Hermitesche Quantorenelimination. Diploma thesis, Universit ät Passau, D-94030 Passau, Germany (2003)

    Google Scholar 

  10. Dolzmann, A.: Solving geometric problems with real quantifier elimination. In: Wang, D., Yang, L., Gao, X.-S. (eds.) ADG 1998. LNCS (LNAI), vol. 1669, pp. 14–29. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  11. Dolzmann, A., Sturm, T.: Simplification of quantifier-free formulae over ordered fields. Journal of Symbolic Computation 24, 209–231 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Weispfenning, V.: Applying quantifier elimination to problems in simulation and optimization. Technical Report MIP-9607, FMI, Universität Passau, D-94030 Passau, Germany (1996); To appear in the Journal of Symbolic Computation

    Google Scholar 

  13. Dolzmann, A., Sturm, T., Weispfenning, V.: A new approach for automatic theorem proving in real geometry. Journal of Automated Reasoning 21, 357–380 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Seidl, A., Sturm, T.: A generic projection operator for partial cylindrical algebraic decomposition. In: Sendra, R. (ed.) Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation (ISSAC 2003), Philadelphia, Pennsylvania, pp. 240–247. ACM Press, New York (2003)

    Chapter  Google Scholar 

  15. Weispfenning, V.: Comprehensive Gröbner bases. Journal of Symbolic Computation 14, 1–29 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Pedersen, P., Roy, M.F., Szpirglas, A.: Counting real zeroes in the multivariate case. In: Eysette, F., Galigo, A. (eds.) Computational Algebraic Geometry. Progress in Mathematics, vol. 109, pp. 203–224. Birkhäuser, Boston (1993); Proceedings of the MEGA 1992 (1992)

    Google Scholar 

  17. Becker, E., Wörmann, T.: On the trace formula for quadratic forms. In: Jacob, W.B., Lam, T.Y., Robson, R.O. (eds.) Recent Advances in Real Algebraic Geometry and Quadratic Forms. Contemporary Mathematics, vol. 155, pp. 271–291. American Mathematical Society, Providence (1994); Proceedings of the RAGSQUAD Year, Berkeley (1990-1991)

    Google Scholar 

  18. Kredel, H., Weispfenning, V.: Computing dimension and independent sets for polynomial ideals. Journal of Symbolic Computation 6, 231–247 (1988); Computational aspects of commutative algebra

    Article  MATH  MathSciNet  Google Scholar 

  19. Chou, S.C.: Mechanical Geometry Theorem Proving. Mathematics and its applications. D. Reidel Publishing Company, Dordrecht (1988)

    MATH  Google Scholar 

  20. Sturm, T.: Real Quantifier Elimination in Geometry. Doctoral dissertation, Department of Mathematics and Computer Science. University of Passau, Germany, D-94030 Passau, Germany (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dolzmann, A., Gilch, L.A. (2004). Generic Hermitian Quantifier Elimination. In: Buchberger, B., Campbell, J. (eds) Artificial Intelligence and Symbolic Computation. AISC 2004. Lecture Notes in Computer Science(), vol 3249. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30210-0_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-30210-0_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-23212-4

  • Online ISBN: 978-3-540-30210-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics