Abstract
Many problems of computational geometry involve non linear polynomials. Many of these problems can be solved with easy algorithms after transforming the polynomial set involved in their specification into Gröbner bases form. Gröbner bases of a system of polynomials are canonical finite sets of multivariate polynomials which define the same algebraic structure as the initial polynomial system. But the computation of Gröbner bases requires a large amount of time and space. However some strategies can be introduced to improve the computation. In this paper, we shortly introduce basic notions of Gröbner bases. Then we propose an algorithm for their computation which is based on a completion procedure for rewrite systems. This algorithm is extended with an orthogonal set of selection strategies which improve each sub-algorithms (reduction, inter reduction, critical pair choice). At last we discuss the use of the strategies applied to some examples (as a restriction of the well known piano mover's problem).
Partially supported by the CHIC ESPRIT Project 5291
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References
F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985.
H. Amet and R. Schott. Déplacement Exact d'un Polygone au Sein d'un Ensemble de Polygones. In 7ème Congrès AFCET Reconnaissance des Formes et Intelligence Artificielle, Paris, pages 749–763, 1989.
J.-C. Latombe. Robot Motion Planning. Kluwer Academic Publishers, Boston/ Dordrecht/ London, 1991.
S. R. Maddila and C. K. Yap. Moving a Polygon arround the Corner in a Corridor. In Proceedings of the ACM Symposium on Computational Geometry, pages 187–182, Yorktown Heights, NY, 1986.
H. Amet, Boissonnat J.-D., and R. Schott. Calcul Dynamique du Diagramme de Voronoï d'un Ensemble de Segments. In Journées de Géométrie Algorithmique, INRIA Sophia-Antipolis (France), pages 143–153, June 1990.
B. Buchberger. Applications of Gröbner Bases in Non-Linear Computational Geometry. In Deepak Kapur and Joseph L. Mundy, editors, Geometric Reasoning, pages 413–446. MIT Press, 1989.
D. E. Knuth and P. B. Bendix. Simple word problems in universal algebras. In J. Leech, editor, Computational Problems in Abstract Algebra, pages 263–297. Oxford, Pergamon Press, 1970.
B. Buchberger. Gröbner Bases: an Algorithmic Method in Polynomial Ideal Theory. In N. K. Bose Ed., editor, Multidimensional Systems theory, pages 184–232. D. Reidel Publishing Company, Dordrecht-Boston-Lancaster, 1985.
B. Buchberger. Some Properties of Gröbner Bases for Polynomial Ideals. J.A.C.M. Sigsam Bull, 10(4):19–24, 1976.
S. R. Czapor. Gröbner Basis Methods for Solving Algebraic Equations. PhD thesis, University of Waterloo, Canada, November 1989.
E. Monfroy. Strategies for Computing Gröbner Bases of a set of polynomials. Internal Report 92-2i, ECRC (European Computer-industry Research Centre), Munich (Germany), 1992.
B. Buchberger. History and Basic Features of the Critical-Pair/Completion Procedure. Journal of Symbolic Computation, 3:3–38, 1987.
F. Winkler. Knuth-Bendix Procedure and Buchberger Algorithm — A Synthesis. In Proceedings of International Symposium on Symbolic and Algebraic Computation (ISSAC'89), pages 55–67, 1989.
B. Buchberger and F. Winkler. Miscellaneous Results on the Construction of Gröbner Bases for Polynomial Ideals. Technical Report 137, Univ. of Linz, Math. Inst., 1979.
W. Boege, R. Gebauer, and H. Kredel. Some Examples for Solving Systems of Algebraic Equations by Calculating Gröbner Bases. Journal of Symbolic Computation, 1:83–98, 1986.
M. Meier, A. Aggoun, D. Chan, P. Dufresne, R. Enders, D. De Villeneuve, A. Herold, P. Kay, B. Perez, E. Van Rossum, and J. Schimpf. SEPIA — an extendible Prolog system. In Proceedings of the 11th World Computer Congress IFIP'89, San Francisco, August 1989.
S. Wolfram. Mathematica: A System for Doing Mathematics by Computer, 1988.
E. Monfroy. Non Linear Constraints: a Language and a Solver. Technical report ECRC-92, ECRC (European Computer-industry Research Centre), Munich (Germany), 1992. To appear.
E. Monfroy. Specification of Geometrical Constraints. Technical report ECRC-92-31, ECRC (European Computer-industry Research Centre), Munich (Germany), 1992.
C. Kirchner and H. Kirchner. Rewriting: Theory and Applications. Working Paper for a D.E.A. lecture at the University of Nancy I, France, 1989.
E. Monfroy. Solutions of Algebraic Equations using Gröbner Bases. Internal Report 92-3i, ECRC (European Computer-industry Research Centre), Munich (Germany), 1992.
E. Monfroy. Applications of Gröbner Bases. Internal Report 92-4i, ECRC (European Computer-industry Research Centre), Munich (Germany), 1992.
E. Monfroy. Spécification Opérationnelle des Contraintes Géométriques: ELIOS. Rapport de DEA CRIN 90-R-130, CRIN Nancy, 1990.
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Monfroy, E. (1993). Gröbner bases: Strategies and applications. In: Calmet, J., Campbell, J.A. (eds) Artificial Intelligence and Symbolic Mathematical Computing. AISMC 1992. Lecture Notes in Computer Science, vol 737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57322-4_9
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DOI: https://doi.org/10.1007/3-540-57322-4_9
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