Abstract
The first part of the paper introduces the decision problem and the quantifier elimination problem for elementary algebra. Tarski's algorithm, which is based on criteria for the existence of solutions of systems of polynomial equations and inequalities generalizing Sturm's theorem, is presented. Then the text is devoted to the cylindrical algebraic decomposition algorithm of Collins, which relies on the basic methods of the theory of semialgebraic sets initiated by Lojasiewicz. Special attention is paid to the topological aspect of the cylindrical algebraic decomposition, which provides a general method (surely too general to be efficient in practice) for robot motion planning. The techniques and results presented here have no pretention to originality. The references given at the end of the paper are just a sample of the literature on the subject, far from being exhaustive.
Preview
Unable to display preview. Download preview PDF.
References
D.S. Arnon, B. Buchberger, ed., Algorithms in real algebraic geometry, Academic Press (1988).
D.S. Arnon, A cluster-based cylindrical algebraic decomposition algorithm, pp. 189–212 in [AB].
D.S. Arnon, G.E. Collins, S. McCallum, Cylindrical algebraic decomposition I: the basic algorithm, SIAM J. Comp 13 (1984), 865–877.
D.S. Arnon, G.E. Collins, S. McCallum, Cylindrical algebraic decomposition II: an adjacency algorithm for the plane, SIAM J. Comp 13 (1984), 878–889.
D.S. Arnon, G.E. Collins, S. McCallum, An adjacency algorithm for cylindrical algebraic decompositions of three-dimensional space, pp. 163–188 in [AB].
D. Arnon, M. Mignotte, On mechanical quantifier elimination for elementary algebra and geometry, pp 237–260 in [AB].
M. Ben-Or, D. Kozen, J. Reif, The complexity of elementary algebra and geometry, J. of Comput. and System Sci. 32 (1986) 251–264.
J. Bochnak, M. Coste, M-F. Roy, Géométrie algébrique réele, Ergebnisse der Math. 12, Springer-Verlag (1987).
G.E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Proc. 2nd GI Conf. Automata Theory & Formal Languages, Springer, Lecture Notes in Computer Science 33 (1985), 134–183.
M. Coste, M-F. Roy, Thom's lemma, the coding of real algebraic numbers and the computation of the topology of semialgebraic sets, pp. 121–130 in [AB].
J.H. Davenport, Computer algebra for cylindric algebraic decomposition, TRITA-NA-8511, NADA, KTH, Stockholm (1985).
J.H. Davenport, A "piano movers" problem, preprint (1985).
J.H. Davenport, J. Heintz, Real quantifier elimination is doubly exponential, pp. 29–36 in [AB].
N. Fitchas, A. Galligo, J. Morgenstern, Algorithmes rapides en séquentiel et en parallèle pour l'élimination de quantificateurs en géométrie élémentaire, Séminaire Structures Algébriques Ordonnées, Univ. Paris VII (1987)
D. Lazard, Quantifier elimination: optimal solution for two classical examples, pp. 261–266 in [AB].
S. Lojasiewicz, Ensembles semi-analytiques, multigraphed, I.H.E.S. (1965).
D.J. Grigoriev, Complexity of deciding Tarski algebra, pp. 65–108 in [AB].
R.G.K. Loos, Generalized polynomial remainder sequences, Computer Algebra — Symbolic and algebraic computation, Springer-Verlag (1982)
J. Marchand, Elimination des quantificateurs et décomposition algébrique cylindrique, in Séminaire "Calcul formel et outils algébriques pour la modélisation géométrique", C.N.R.S. (1988).
S. McCallum, An improved projection operation for cylindric algebraic decomposition, Computer Science Tech. Report 548, Univ. Wisconsin at Madison (1985), see also pp. 141–162 in [AB].
A. Paugam, Comparaison entre 3 algorithmes d'élimination des quantificateurs sur les corps réels clos, Thèse, Univ. Rennes 1 (1986).
J-J. Risler, About the piano mover's problem, preprint (1987).
J. Schwartz, M. Sharir, On the "piano movers" problem II. General techniques for computing topological properties of real algebraic manifolds, Advances in Applied Mathematics 4 (1983), 298–351.
A. Seidenberg, A new decision method for elementary algebra, Annals of Math. 60 (1954) 365–371.
A. Tarski, A decision method for elementary algebra and geometry, 2nd ed., Univ. Calif. Press, Berkeley (1951).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Coste, M. (1989). Effective semialgebraic geometry. In: Boissonnat, J.D., Laumond, J.P. (eds) Geometry and Robotics. GeoRob 1988. Lecture Notes in Computer Science, vol 391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51683-2_21
Download citation
DOI: https://doi.org/10.1007/3-540-51683-2_21
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51683-5
Online ISBN: 978-3-540-46748-9
eBook Packages: Springer Book Archive