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

  • Fabrice Rouillier
  • Mohab Safey El Din
  • Éric Schost
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2061)

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
  2. 2.
  3. 3.
  4. 4.
  5. 5.
    P. Aubry, Ensembles Triangulaires de Polynômes et Résolution de Systémes Algébriques, Implantations en Axiom, PhD thesis, Université Paris VI, 1999.Google Scholar
  6. 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. 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. 8.
    B. Bank, M. Giusti, J. Heintz, M. Mbakop, Polar Varieties and Efficient Real Elimination, to appear in Mathematische Zeitschrift (2000).Google Scholar
  9. 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. 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.MathSciNetGoogle Scholar
  11. 11.
    G. E. Collins, H. Hong,Partial Cylindrical Algebraic Decomposition, Journal of Symbolic Computation, Vol. 12, No. 3, pages 299–328, 1991.MATHMathSciNetCrossRefGoogle Scholar
  12. 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. 13.
    P. Conti, C. Traverso, Algorithms for the Real Radical, Unpublished manuscript.Google Scholar
  14. 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. 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. 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. 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. 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. 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. 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. 21.
    M. Kalkbrener, Three Contributions to Elimination Theory, PhD thesis, RISCLinz, Johannes Kepler University, 1991.Google Scholar
  22. 22.
    L. Kronecker,Grundzüge einer arithmetischen Theorie der algebraischen Größen, Journal Reine Angew. Mathematik, Vol. 92, pages 1–122, 1882.Google Scholar
  23. 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. 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. 25.
    F. Rouillier, Algorithmes Efficaces pour l’ Étude des Zéros Réels des Systémes Polynomiaux, PhD thesis, Université de Rennes I, 1996.Google Scholar
  26. 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.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    R. Rioboo, Computing with Infinitesimals, Manuscript.Google Scholar
  28. 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.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    F. Rouillier, P. Zimmermann, Efficient Isolation of a Polynomial Real Roots, Research Report, INRIA, No. RR-4113, 2001.Google Scholar
  30. 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. 31.
    M. Safey El Din, Résolution Réelle des Systèmes Polynomiaux en Dimension Positive, PhD thesis, Université Paris VI, 2001.Google Scholar
  32. 32.
    É. Schost, Computing Parametric Geometric Resolutions, Preprint, École Polytechnique, 2000.Google Scholar
  33. 33.
    É. Schost, Sur la Résolution des Systèmes Polynomiaux á Paramètres, PhD thesis, École Polytechnique, 2000.Google Scholar
  34. 34.
    D. Wang, Computing Triangular Systems and Regular Systems, Journal of Symbolic Computation, Vol. 30, No. 2, pages 221–236, 2000.MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    J. Von Zur Gathen, J. Gerhardt, Modern Computer Algebra, Cambridge University Press, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Fabrice Rouillier
    • 1
  • Mohab Safey El Din
    • 2
  • Éric Schost
    • 3
  1. 1.LORIAINRIA-LorraineNancyFrance
  2. 2.CALFOR, LIP6Université Paris VIParisFrance
  3. 3.Laboratoire GAGEÉcole PolytechniquePalaiseauFrance

Personalised recommendations