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Betti number bounds for fewnomial hypersurfaces via stratified Morse theory

Authors: Frédéric Bihan and Frank Sottile
Journal: Proc. Amer. Math. Soc. 137 (2009), 2825-2833
MSC (2000): Primary 14P25
Published electronically: April 23, 2009
MathSciNet review: 2506438
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Abstract: We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a fewnomial hypersurface in $ \mathbb{R}^N_{>}$.

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  • 1. R. Benedetti and J.-J. Risler, Real algebraic and semi-algebraic sets, Actualités Mathématiques, Hermann, Paris, 1990. MR 1070358 (91j:14045)
  • 2. F. Bihan, J.M. Rojas, and F. Sottile, Sharpness of fewnomial bounds and the number of components of a fewnomial hypersurface, Algorithms in Algebraic Geometry (A. Dickenstein, F. Schreyer, and A. Sommese, eds.), IMA Volumes in Mathematics and its Applications, vol. 146, Springer, 2008, pp. 15-20. MR 2397935
  • 3. F. Bihan and F. Sottile, New fewnomial upper bounds from Gale dual polynomial systems, Moscow Mathematical Journal 7 (2007), no. 3, 387-407. MR 2343138 (2008g:13038)
  • 4. J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36, Springer-Verlag, Berlin, 1998, revised translation of the 1987 French original. MR 949442 (90b:14030)
  • 5. M. Goresky and R. MacPherson, Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 14, Springer-Verlag, Berlin, 1988. MR 932724 (90d:57039)
  • 6. A.G. Khovanskii, Fewnomials, Trans. of Math. Monographs, 88, Amer. Math. Soc, Providence, RI, 1991. MR 1108621 (92h:14039)
  • 7. J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275-280. MR 0161339 (28:4547)
  • 8. O. A. Oleĭnik, Estimates of the Betti numbers of real algebraic hypersurfaces, Mat. Sbornik N.S. 28(70) (1951), 635-640. MR 0044864 (13:489b)
  • 9. D. Perrucci, Some bounds for the number of components of real zero sets of sparse polynomials, Discrete Comput. Geom. 34 (2005), no. 3, 475-495. MR 2160050 (2006j:12001)
  • 10. J.-J. Risler, Les nombres de Betti des ensembles algébriques réels, Gaz. Math. No. 54 (1992), 57-58.
  • 11. J. Maurice Rojas, Some speed-ups and speed limits for real algebraic geometry, J. Complexity 16 (2000), no. 3, 552-571. MR 1787885 (2001k:14103)
  • 12. R. Thom, Sur l'homologie des variétés algébriques réelles, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, NJ, 1965, pp. 255-265. MR 0200942 (34:828)

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Additional Information

Frédéric Bihan
Affiliation: Laboratoire de Mathématiques, Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France

Frank Sottile
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Keywords: Stratified Morse theory, fewnomials, Betti numbers
Received by editor(s): June 19, 2008
Published electronically: April 23, 2009
Additional Notes: The second author was supported by NSF CAREER grant DMS-0538734 and NSF grant DMS-0701050
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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