The number of roots of a lacunary bivariate polynomial on a line

We prove that a polynomial f ∈ R [x, y] with t non-zero terms, restricted to a real line y = a x + b, either has at most 6 t - 4 zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y - a x - b ∈ K [x, y] divides a lacun...

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Detalhes bibliográficos
Autor: Avendaño, M.
Formato: artículo
Estado:Versión publicada
Fecha de publicación:2009
País:Argentina
Recursos:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
Repositorio:Biblioteca Digital (UBA-FCEN)
Idioma:inglés
OAI Identifier:paperaa:paper_07477171_v44_n9_p1280_Avendano
Acesso em linha:http://hdl.handle.net/20.500.12110/paper_07477171_v44_n9_p1280_Avendano
Access Level:acceso abierto
Palavra-chave:Descartes' rule of signs
Factorization of polynomials
Fewnomials
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spelling The number of roots of a lacunary bivariate polynomial on a lineAvendaño, M.Descartes' rule of signsFactorization of polynomialsFewnomialsWe prove that a polynomial f ∈ R [x, y] with t non-zero terms, restricted to a real line y = a x + b, either has at most 6 t - 4 zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y - a x - b ∈ K [x, y] divides a lacunary polynomial f ∈ K [x, y], where K is a real number field. The number of bit operations performed by the algorithm is polynomial in the number of non-zero terms of f, in the logarithm of the degree of f, in the degree of the extension K / Q and in the logarithmic height of a, b and f. © 2009 Elsevier Ltd. All rights reserved.Fil:Avendaño, M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.2009info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfhttp://hdl.handle.net/20.500.12110/paper_07477171_v44_n9_p1280_AvendanoJ. Symb. Comput. 2009;44(9):1280-1284reponame:Biblioteca Digital (UBA-FCEN)instname:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesinstacron:UBA-FCENenginfo:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by/2.5/ar2024-05-10T10:42:46Zpaperaa:paper_07477171_v44_n9_p1280_AvendanoInstitucionalhttps://digital.bl.fcen.uba.ar/Universidad públicaNo correspondehttps://digital.bl.fcen.uba.ar/cgi-bin/oaiserver.cgiana@bl.fcen.uba.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:18962024-05-10 10:42:47.905Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesfalse
dc.title.none.fl_str_mv The number of roots of a lacunary bivariate polynomial on a line
title The number of roots of a lacunary bivariate polynomial on a line
spellingShingle The number of roots of a lacunary bivariate polynomial on a line
Avendaño, M.
Descartes' rule of signs
Factorization of polynomials
Fewnomials
title_short The number of roots of a lacunary bivariate polynomial on a line
title_full The number of roots of a lacunary bivariate polynomial on a line
title_fullStr The number of roots of a lacunary bivariate polynomial on a line
title_full_unstemmed The number of roots of a lacunary bivariate polynomial on a line
title_sort The number of roots of a lacunary bivariate polynomial on a line
dc.creator.none.fl_str_mv Avendaño, M.
author Avendaño, M.
author_facet Avendaño, M.
author_role author
dc.subject.none.fl_str_mv Descartes' rule of signs
Factorization of polynomials
Fewnomials
topic Descartes' rule of signs
Factorization of polynomials
Fewnomials
description We prove that a polynomial f ∈ R [x, y] with t non-zero terms, restricted to a real line y = a x + b, either has at most 6 t - 4 zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y - a x - b ∈ K [x, y] divides a lacunary polynomial f ∈ K [x, y], where K is a real number field. The number of bit operations performed by the algorithm is polynomial in the number of non-zero terms of f, in the logarithm of the degree of f, in the degree of the extension K / Q and in the logarithmic height of a, b and f. © 2009 Elsevier Ltd. All rights reserved.
publishDate 2009
dc.date.none.fl_str_mv 2009
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.12110/paper_07477171_v44_n9_p1280_Avendano
url http://hdl.handle.net/20.500.12110/paper_07477171_v44_n9_p1280_Avendano
dc.language.none.fl_str_mv eng
language eng
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by/2.5/ar
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by/2.5/ar
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv J. Symb. Comput. 2009;44(9):1280-1284
reponame:Biblioteca Digital (UBA-FCEN)
instname:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
instacron:UBA-FCEN
instname_str Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
instacron_str UBA-FCEN
institution UBA-FCEN
reponame_str Biblioteca Digital (UBA-FCEN)
collection Biblioteca Digital (UBA-FCEN)
repository.name.fl_str_mv Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
repository.mail.fl_str_mv ana@bl.fcen.uba.ar
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