Abstract
For every polynomial mapf=(f 1,…,f k): ℝn→ℝk, we consider the number of connected components of its zero set,B(Z f) and two natural “measures of the complexity off,” that is the triple(n, k, d), d being equal to max(degree off i), and thek-tuple (Δ1,...,Δ4), Δ k being the Newton polyhedron off i respectively. Our aim is to boundB(Z f) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom’s bound μ d (n)=d(2d−1)n−1. Considered as a polynomial ind, μ d (n) has leading coefficient equal to 2n−1. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μ d (n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)n k−1dn, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument, Smith's theory, and information about the sum of Betti numbers of complex complete intersections.
Article PDF
Similar content being viewed by others
References
[BR] Benedetti, R., and Risler J. J.,Real Algebraic and Semi-Algebraic Set, Hermann, Paris, 1990.
[B1] Ben-Or, M., Lower bounds for algebraic computation trees,Proc. 15th ACM Symp. on Theory of Computing, pp. 80–86, 1983.
[BCR] Bochnak, J., Coste, M., and Roy, M. F.,Géometrie algébrique réelle, Springer-Verlag, Berlin, 1987.
[B2] Bredon, G. F.,Introduction to Compact Transformation Groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York, 1972.
[DH] Danilov, V. I., and Hovansky, A. G., Newton polyhedra and an algorithm for computing Hodge Deligne numbers,Math. USSR-Izv.,29, 279–298, 1987.
[D] Durfee, A. H., Neighbourhoods of algebraic sets,Trans. Amer. Math. Soc.,276, 517–530, 1983.
[F] Fary, I., Cohomologie des varietes algébriques,Ann. of Math.,65, 21–73, 1957
[GWDL] Gibson, C. G., Wirthmuller, K., du Plessis, A. A., and Looijenga, E. J. N.,Topological Stability of Smooth Mappings, Lectures Notes in Mathematics, Vol. 552. Springer-Verlag, Berlin, 1976.
[GP] Goodman, J. E., and Pollack, R., Upper bounds for configurations and polytopes in ℝ d ,Discrete Comput. Geom.,1, 219–227, 1986.
[H1] Hovansky, A. G., Newton polyhedra and toric varieties,Functional Anal. Appl.,11, 289–296, 1977.
[H2] Hovansky, A. G., Newton polyhedra and the genus of complete intersections,Functional Anal. Appl.,12, 38–46, 1978.
[M1] Milnor, J., On the Betti numbers of real varieties,Proc. Amer. Math. Soc.,15, 275–280, 1964.
[M2] Milnor, J.,Singular Points of Complex Hypersurfaces, Annals of Mathematical Studies, Vol. 61, Princeton University Press, Princeton, NJ, 1968.
[R1] Risler, J. J., Complexité et géometrie réele,sem Bourbaki Asterisque 133/134 89–100, 1986.
[R2] Rokhlin, V. A., Congruences mod 16 in Hilbert's 16th problem,Functional Anal. Appl.,6, 301–306, 1972.
[T] Thom, R., Sur l'homologie des varietes algébriques réelles, inDifferential and Combinatorical Topology, pp. 255–265 (Cairns, S., ed.), Princeton University Press, Princeton, NJ, 1965.
[W] Warren, E. H., Lower bounds for approximation by nonlinear manifolds,Trans. Amer. Math Soc.,133, 167–178, 1968.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Benedetti, R., Loeser, F. & Risler, J.J. Bounding the number of connected components of a real algebraic set. Discrete Comput Geom 6, 191–209 (1991). https://doi.org/10.1007/BF02574685
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02574685