Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-30T19:58:41.206Z Has data issue: false hasContentIssue false

On the number of cells defined by a family of polynomials on a variety

Published online by Cambridge University Press:  26 February 2010

Saugata Basu
Affiliation:
Courant Institute of Mathematical Science, New York University, New York, NY 10012, U.S.A.
Richard Pollak
Affiliation:
Courant Institute of Mathematical Science, New York University, New York, NY 10012, U.S.A.
Marie-Françoise Roy
Affiliation:
IRMAR (URA CNRS 305), Université de Rennes, Campus de Beaulieu, 35042 Rennes cedex, France.
Get access

Abstract

Let R be a real closed field and a variety of real dimension k′ which is the zero set of a polynomial QR[X1,…, Xk] of degree at most d. Given a family of s polynomials = {P1,…, Ps}⊂R[X1,…,Xk] where each polynomial in has degree at most d, we prove that the number of cells defined by over is (O(d))k Note that the combinatorial part of the bound depends on the dimension of the variety rather than on the dimension of the ambient space.

Type
Research Article
Copyright
Copyright © University College London 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Basu, S., Pollack, R. and Roy, M.-F.. A new algorithm to find a point in every cell denned by a family of polynomials. In Quantifier Elimination and Cylindrical Algebraic Decomposition, edited by Caviness, B. and Johnson, J. (Springer-Verlag) to appear.Google Scholar
2.Basu, S., Pollack, R. and Roy, M.-F.. Computing points meeting every cell on a variety. In The Algorithmic Foundations of Robotics, edited by Goldberg, K., Halperin, D., Latombe, J. C. and Wilson, R. (A. K. Peters, Boston, MA, 1955), 537555.Google Scholar
3.Basu, S., Pollack, R. and Roy, M.-F.. On the combinatorial and algebraic complexity of Quantifier Elimination. In Proc. 35th Annual IEEE Sympos. on the Foundations of Computer Science, 632641 (1994).Google Scholar
4.Bochnak, J., Coste, M. and Roy, M.-F.. Géometrie algébrique réelle (Springer-Verlag, 1987).Google Scholar
5.Canny, J.. Some Practical Tools for Algebraic Geometry. In Technical report in Spring school on robot motion planning (PROMOTION ESPRIT, 1993).Google Scholar
6.Canny, J.. Computing road maps in general semi-algebraic sets. The Computer Journal, 36(1993), 504514.Google Scholar
7.Canny, J.. Improved algorithms for sign determination and existential quantifier elimination. The Computer Journal, 36 (1993), 409418.Google Scholar
8.Edelsbrunner, H.. Algorithms in Combinatorial Geometry (Springer-Verlag, Berlin, 1987).Google Scholar
9.Goodman, J. E., Pollack, R. and Wenger, R.. Bounding the number of geometric permutations induced by k-transversals. In Proc. 10th Ann. ACM Sympos. Comput. Geom. (1994), 192197.Google Scholar
10.Heintz, J., Roy, M.-F. and Solernó, P.. On the complexity of semi-algebraic sets. In Proc. IFIP San Francisco (North-Holland, 1989), 293298.Google Scholar
11.Milnor, J.. On the Betti numbers of real varieties. Proc. Amer. Math. Soc, 15 (1964), 275280.Google Scholar
12.Olelnik, O. A.. Estimates of the Betti numbers of real algebraic hypersurfaces (Russian). Mat. Sb. (N.S.), 28 (70) (1951), 635640.Google Scholar
13.Petrovsky, I. G. and Olelnik, O. A.. On the topology of real algebraic surfaces. Izvestiaya Akademii Nauk SSSR. Serija Matematičeskaya, 13 (1949), 389402.Google Scholar
14.Pollack, R. and Roy, M.-F.. On the number of cells defined by a set of polynomials. C. R. Acad. Sci. Paris, 316 (1993), 573577.Google Scholar
15.Renegar, J.. On the computational complexity and geometry of the first order theory of the reals. J. of Symbolic Comput., 13 (1992), 255352.Google Scholar
16.Thom, R.. Sur l'homologie des variétés algébriques réelles. In Differential and Combinatorial Topology (Princeton University Press, Princeton, 1965), 255265.CrossRefGoogle Scholar