Artículo
Abstract:
We prove that the zero set of a 4-nomial in n variables in the positive orthant has at most three connected components. This bound, which does not depend on the degree of the polynomial, not only improves the best previously known bound (which was 10) but is optimal as well. In the general case we prove that the number of connected components of the zero set of an m-nomial in n variables in the positive orthant is lower than or equal to (n+1) m-121 + (m - 1)(m - 2)/2, improving slightly the known bounds. Finally, we show that for generic exponents, the number of non-compact connected components of the zero set of a 5-nomial in three variables in the positive octant is at most 12. This strongly improves the best previously known bound, which was 10,384. All the bounds obtained in this paper continue to hold for real exponents. © 2005 Springer Science+Business Media, Inc.
Registro:
| Documento: | Artículo |
| Título: | Some bounds for the number of components of real zero sets of sparse polynomials |
| Autor: | Perrucci, D. |
| Filiación: | Departamento de Matematica, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pab. I (1428), Buenos Aires, Argentina |
| Año: | 2005 |
| Volumen: | 34 |
| Número: | 3 |
| Página de inicio: | 475 |
| Página de fin: | 495 |
| DOI: | http://dx.doi.org/10.1007/s00454-005-1179-x |
| Título revista: | Discrete and Computational Geometry |
| Título revista abreviado: | Discrete Comput. Geom. |
| ISSN: | 01795376 |
| PDF: | https://bibliotecadigital.exactas.uba.ar/download/paper/paper_01795376_v34_n3_p475_Perrucci.pdf |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01795376_v34_n3_p475_Perrucci |
Referencias:
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- Gabrielov, A., Zell, T., On the number of connected components of the relative closure of a semi-Pfaffian family (2003) Algorithmic and Quantitative Real Algebraic Geometry, pp. 65-75. , AMS, Providence, RI
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- (1980) Soviet Math. Dokl., 22 (3), pp. 762-765
- Khovanski, A.G., Fewnomials (1991) Translations of Mathematical Monographs, 88. , AMS, Providence, RI
- Lefschetz, S., (1945) Intmduction to Topology, , Princeton University Press, Princeton, NJ
- Li, T., Rojas, M., Wang, X., Counting real connected components of trinomial curve intersections and m-nomial hypersurfaces (2003) Discrete Comput. Geom., 30 (3), pp. 379-414
- Milnor, J.W., (1965) Topology from the Differentiable Viewpoint, , Princeton University Press, Princeton, NJ
- Voorhoeve, M., A generalization of Descartes' rule (1979) J. London Math. Soc. (2), 20 (3), pp. 446-456
Citas:
---------- APA ----------
(2005) . Some bounds for the number of components of real zero sets of sparse polynomials. Discrete and Computational Geometry, 34(3), 475-495.http://dx.doi.org/10.1007/s00454-005-1179-x
---------- CHICAGO ----------
Perrucci, D. "Some bounds for the number of components of real zero sets of sparse polynomials" . Discrete and Computational Geometry 34, no. 3 (2005) : 475-495.http://dx.doi.org/10.1007/s00454-005-1179-x
---------- MLA ----------
Perrucci, D. "Some bounds for the number of components of real zero sets of sparse polynomials" . Discrete and Computational Geometry, vol. 34, no. 3, 2005, pp. 475-495.http://dx.doi.org/10.1007/s00454-005-1179-x
---------- VANCOUVER ----------
Perrucci, D. Some bounds for the number of components of real zero sets of sparse polynomials. Discrete Comput. Geom. 2005;34(3):475-495.http://dx.doi.org/10.1007/s00454-005-1179-x
