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.
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References
P. Aubry, Ensembles Triangulaires de Polynômes et Résolution de Systémes Algébriques, Implantations en Axiom, PhD thesis, Université Paris VI, 1999.
P. Aubry, F. Rouillier, M. Safey El Din, Real Solving for Positive Dimensional Systems, Research Report, Laboratoire d’Informatique de Paris VI, March 2000.
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.
B. Bank, M. Giusti, J. Heintz, M. Mbakop, Polar Varieties and Efficient Real Elimination, to appear in Mathematische Zeitschrift (2000).
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.
E. Becker, R. Neuhaus, Computation of Real Radicals for Polynomial Ideals, Computational Algebraic Geometry, Progress in Math., Vol. 109, pages 1–20, Birkhäuser, 1993.
G. E. Collins, H. Hong,Partial Cylindrical Algebraic Decomposition, Journal of Symbolic Computation, Vol. 12, No. 3, pages 299–328, 1991.
G. E. Collins, Quantifier Elimination for Real Closed Field by Cylindrical Algebraic Decomposition, Lectures Notes in Computer Science, Vol. 33, pages 515–532, 1975.
P. Conti, C. Traverso, Algorithms for the Real Radical, Unpublished manuscript.
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.
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.
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.
L. Gonzalez-Vega, Applying Quantifier Elimination to the Birkhoff Interpolation Problem, Journal of Symbolic Computation Vol. 22, No. 1, pages 83–103, 1996.
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.
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.
H. Hong, Comparison of Several Decision Algorithms for the Existential Theory of the Reals, Research Report, RISC-Linz, Johannes Kepler University, 1991.
M. Kalkbrener, Three Contributions to Elimination Theory, PhD thesis, RISCLinz, Johannes Kepler University, 1991.
L. Kronecker,Grundzüge einer arithmetischen Theorie der algebraischen Größen, Journal Reine Angew. Mathematik, Vol. 92, pages 1–122, 1882.
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.
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.
F. Rouillier, Algorithmes Efficaces pour l’ Étude des Zéros Réels des Systémes Polynomiaux, PhD thesis, Université de Rennes I, 1996.
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.
R. Rioboo, Computing with Infinitesimals, Manuscript.
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.
F. Rouillier, P. Zimmermann, Efficient Isolation of a Polynomial Real Roots, Research Report, INRIA, No. RR-4113, 2001.
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.
M. Safey El Din, Résolution Réelle des Systèmes Polynomiaux en Dimension Positive, PhD thesis, Université Paris VI, 2001.
É. Schost, Computing Parametric Geometric Resolutions, Preprint, École Polytechnique, 2000.
É. Schost, Sur la Résolution des Systèmes Polynomiaux á Paramètres, PhD thesis, École Polytechnique, 2000.
D. Wang, Computing Triangular Systems and Regular Systems, Journal of Symbolic Computation, Vol. 30, No. 2, pages 221–236, 2000.
J. Von Zur Gathen, J. Gerhardt, Modern Computer Algebra, Cambridge University Press, 1999.
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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
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DOI: https://doi.org/10.1007/3-540-45410-1_3
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