Skip to main content
SpringerLink
Log in

Solving the Birkhoff Interpolation Problem via the Critical Point Method: An Experimental Study

  • Conference paper
  • First Online:
Automated Deduction in Geometry (ADG 2000)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2061))

Included in the following conference series:

Abstract

Following the work of Gonzalez-Vega, this paper is devoted to showing how to use recent algorithmic tools of computational real algebraic geometry to solve the Birkhoff Interpolation Problem. We recall and partly improve two algorithms to find at least one point in each connected component of a real algebraic set defined by a single equation or a system of polynomial equations, both based on the computation of the critical points of a distance function.

These algorithms are used to solve the Birkhoff Interpolation Problem in a case which was known as an open problem. The solution is available at the U.R.L.: http://www-calfor.lip6.fr/~safey/applications.html.

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. http://www-calfor.lip6.fr/~safey/applications.html

  2. http://www.medicis.polytechnique.fr

  3. http://www.tera.medicis.polytechnique.fr

  4. http://www-calfor.lip6.fr/~jcf

  5. P. Aubry, Ensembles Triangulaires de Polynômes et Résolution de Systémes Algébriques, Implantations en Axiom, PhD thesis, Université Paris VI, 1999.

    Google Scholar 

  6. P. Aubry, F. Rouillier, M. Safey El Din, Real Solving for Positive Dimensional Systems, Research Report, Laboratoire d’Informatique de Paris VI, March 2000.

    Google Scholar 

  7. B. Bank, M. Giusti, J. Heintz, M. Mbakop, Polar Varieties and Efficient Real Equation Solving, Journal of Complexity, Vol. 13, pages 5–27, 1997; best paper award 1997.

    Google Scholar 

  8. B. Bank, M. Giusti, J. Heintz, M. Mbakop, Polar Varieties and Efficient Real Elimination, to appear in Mathematische Zeitschrift (2000).

    Google Scholar 

  9. S. Basu, R. Pollack, M.-F. Roy, On the Combinatorial and Algebraic Complexity of Quantifier Elimination. Journal of the Association for Computing Machinery, Vol. 43, pages 1002–1045, 1996.

    Google Scholar 

  10. E. Becker, R. Neuhaus, Computation of Real Radicals for Polynomial Ideals, Computational Algebraic Geometry, Progress in Math., Vol. 109, pages 1–20, Birkhäuser, 1993.

    MathSciNet  Google Scholar 

  11. G. E. Collins, H. Hong,Partial Cylindrical Algebraic Decomposition, Journal of Symbolic Computation, Vol. 12, No. 3, pages 299–328, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. E. Collins, Quantifier Elimination for Real Closed Field by Cylindrical Algebraic Decomposition, Lectures Notes in Computer Science, Vol. 33, pages 515–532, 1975.

    Google Scholar 

  13. P. Conti, C. Traverso, Algorithms for the Real Radical, Unpublished manuscript.

    Google Scholar 

  14. M. Giusti, J. Heintz, La D’etermination des Points Isol’es et de la Dimension d’une Variátá Algábrique Ráelle peut se faire en Temps Polynomial, Computational Algebraic Geometry and Commutative Algebra, Symposia Matematica, Vol. 34, D. Eisenbud and L. Robbiano (eds.), pages 216–256, Cambridge University Press, 1993.

    Google Scholar 

  15. M. Giusti, G. Lecerf, B. Salvy, A Gröbner Free Alternative for Solving Polynomial Systems, Journal of Complexity, Vol. 17, No. 1, pages 154–211, 2001.

    Google Scholar 

  16. D. Grigor’ev, N. Vorobjov, Solving Systems of Polynomial Inequalities in Subexponential Time, Journal of Symbolic Computation, Vol. 5, No. 1-2, pages 37–64, 1988.

    Google Scholar 

  17. L. Gonzalez-Vega, Applying Quantifier Elimination to the Birkhoff Interpolation Problem, Journal of Symbolic Computation Vol. 22, No. 1, pages 83–103, 1996.

    Google Scholar 

  18. M.-J. Gonzalez-Lopez, L. Gonzalez-Vega, Project 2: The Birkhoff Interpolation Problem, Some Tapas of Computer Algebra, A. Cohen (ed.), pages 297–310, Springer, 1999.

    Google Scholar 

  19. J. Heintz, M.-F. Roy, P. Solerno, On the Theoretical and Practical Complexity of the Existential Theory of the Reals, The Computer Journal, Vol.36, No.5, pages 427–431, 1993.

    Google Scholar 

  20. H. Hong, Comparison of Several Decision Algorithms for the Existential Theory of the Reals, Research Report, RISC-Linz, Johannes Kepler University, 1991.

    Google Scholar 

  21. M. Kalkbrener, Three Contributions to Elimination Theory, PhD thesis, RISCLinz, Johannes Kepler University, 1991.

    Google Scholar 

  22. L. Kronecker,Grundzüge einer arithmetischen Theorie der algebraischen Größen, Journal Reine Angew. Mathematik, Vol. 92, pages 1–122, 1882.

    Google Scholar 

  23. M. Moreno Maza, Calculs de Pgcd au-dessus des Tours d’Extensions Simples et Résolution des Systémes d’Equations Algébriques, PhD thesis, Université Paris VI, 1997.

    Google Scholar 

  24. J. Renegar, On the Computational Complexity and Geometry of the First Order Theory of the Reals, Journal of Symbolic Computation, Vol.13, No.3, pages 255–352, 1992.

    Google Scholar 

  25. F. Rouillier, Algorithmes Efficaces pour l’ Étude des Zéros Réels des Systémes Polynomiaux, PhD thesis, Université de Rennes I, 1996.

    Google Scholar 

  26. F. Rouillier, Solving Zero-Dimensional Systems through the Rational Univariate Representation, Applicable Algebra in Engineering Communications and Computing, Vol.9, No.5, pages 433–461, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  27. R. Rioboo, Computing with Infinitesimals, Manuscript.

    Google Scholar 

  28. F. Rouillier, M.-F. Roy, M. Safey El Din, Finding at Least One Point in Each Connected Component of a Real Algebraic Set Defined by a Single Equation, Journal of Complexity, Vol. 16, No. 4, pages 716–750, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  29. F. Rouillier, P. Zimmermann, Efficient Isolation of a Polynomial Real Roots, Research Report, INRIA, No. RR-4113, 2001.

    Google Scholar 

  30. M.-F. Roy, Basic Algorithms in Real Algebraic Geometry: From Sturm Theorem to the Existential Theory of Reals, Lectures on Real Geometry in memoriam of Mario Raimondo, Expositions in Mathematics, Vol. 23, pages 1–67, Berlin, 1996.

    Google Scholar 

  31. M. Safey El Din, Résolution Réelle des Systèmes Polynomiaux en Dimension Positive, PhD thesis, Université Paris VI, 2001.

    Google Scholar 

  32. É. Schost, Computing Parametric Geometric Resolutions, Preprint, École Polytechnique, 2000.

    Google Scholar 

  33. É. Schost, Sur la Résolution des Systèmes Polynomiaux á Paramètres, PhD thesis, École Polytechnique, 2000.

    Google Scholar 

  34. D. Wang, Computing Triangular Systems and Regular Systems, Journal of Symbolic Computation, Vol. 30, No. 2, pages 221–236, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  35. J. Von Zur Gathen, J. Gerhardt, Modern Computer Algebra, Cambridge University Press, 1999.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LORIA, INRIA-Lorraine, Nancy, France

    Fabrice Rouillier

  2. CALFOR, LIP6, Université Paris VI, Paris, France

    Mohab Safey El Din

  3. Laboratoire GAGE, École Polytechnique, Palaiseau, France

    Éric Schost

Authors
  1. Fabrice Rouillier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohab Safey El Din
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Éric Schost
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Zentrum Mathematik, SB4, Technische Universität München, 80290, München, Germany

    Jürgen Richter-Gebert

  2. Laboratoire d’Informatique de Paris 6, Université Pierre et Marie Curie — CNRS, 4 place Jussieu, 75252, Paris Cedex 05, France

    Dongming Wang

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Rouillier, F., El Din, M.S., Schost, É. (2001). Solving the Birkhoff Interpolation Problem via the Critical Point Method: An Experimental Study. In: Richter-Gebert, J., Wang, D. (eds) Automated Deduction in Geometry. ADG 2000. Lecture Notes in Computer Science(), vol 2061. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45410-1_3

Download citation

  • DOI: https://doi.org/10.1007/3-540-45410-1_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42598-4

  • Online ISBN: 978-3-540-45410-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation