Some bounds for the number of components of real zero sets of sparse polynomials

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 p...

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Autor principal: Perrucci, Daniel
Publicado: 2005
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01795376_v34_n3_p475_Perrucci
http://hdl.handle.net/20.500.12110/paper_01795376_v34_n3_p475_Perrucci
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_01795376_v34_n3_p475_Perrucci
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spelling paper:paper_01795376_v34_n3_p475_Perrucci2023-06-08T15:19:28Z Some bounds for the number of components of real zero sets of sparse polynomials Perrucci, Daniel 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. Fil:Perrucci, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2005 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01795376_v34_n3_p475_Perrucci http://hdl.handle.net/20.500.12110/paper_01795376_v34_n3_p475_Perrucci
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description 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.
author Perrucci, Daniel
spellingShingle Perrucci, Daniel
Some bounds for the number of components of real zero sets of sparse polynomials
author_facet Perrucci, Daniel
author_sort Perrucci, Daniel
title Some bounds for the number of components of real zero sets of sparse polynomials
title_short Some bounds for the number of components of real zero sets of sparse polynomials
title_full Some bounds for the number of components of real zero sets of sparse polynomials
title_fullStr Some bounds for the number of components of real zero sets of sparse polynomials
title_full_unstemmed Some bounds for the number of components of real zero sets of sparse polynomials
title_sort some bounds for the number of components of real zero sets of sparse polynomials
publishDate 2005
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01795376_v34_n3_p475_Perrucci
http://hdl.handle.net/20.500.12110/paper_01795376_v34_n3_p475_Perrucci
work_keys_str_mv AT perruccidaniel someboundsforthenumberofcomponentsofrealzerosetsofsparsepolynomials
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