Sharp bounds for the number of roots of univariate fewnomials

Autores
Avendaño, Martín; Krick, Teresa Elena Genoveva
Año de publicación
2011
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Let K be a field and t ≥ 0. Denote by B m ( t, K ) the supremum of the number of roots in K ∗ , counted with multiplicities, that can have a non-zero polynomial in K [ x ] with at most t + 1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that B m ( t, L ) ≤ t 2 B m ( t, K ) for any local field L with a non-archimedean valuation v : L → R ∪{∞} such that v | Z 6 =0 ≡ 0 and residue field K , and that B m ( t, K ) ≤ ( t 2 − t +1)( p f − 1) for any finite extension K/ Q p with residual class degree f and ramification index e , assuming that p > t + e . For any finite extension K/ Q p , for p odd, we also show the lower bound B m ( t, K ) ≥ (2 t − 1)( p f − 1), which gives the sharp estimation B m (2 , K ) = 3( p f − 1) for trinomials when p > 2 + e .
Fil: Avendaño, Martín. Texas A&M University; Estados Unidos
Fil: Krick, Teresa Elena Genoveva. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Materia
Lacunary Polynomials
Root Counting
Local Fields
Generealized Vandermonde Determinants
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/14919

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spelling Sharp bounds for the number of roots of univariate fewnomialsAvendaño, MartínKrick, Teresa Elena GenovevaLacunary PolynomialsRoot CountingLocal FieldsGenerealized Vandermonde Determinantshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Let K be a field and t ≥ 0. Denote by B m ( t, K ) the supremum of the number of roots in K ∗ , counted with multiplicities, that can have a non-zero polynomial in K [ x ] with at most t + 1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that B m ( t, L ) ≤ t 2 B m ( t, K ) for any local field L with a non-archimedean valuation v : L → R ∪{∞} such that v | Z 6 =0 ≡ 0 and residue field K , and that B m ( t, K ) ≤ ( t 2 − t +1)( p f − 1) for any finite extension K/ Q p with residual class degree f and ramification index e , assuming that p > t + e . For any finite extension K/ Q p , for p odd, we also show the lower bound B m ( t, K ) ≥ (2 t − 1)( p f − 1), which gives the sharp estimation B m (2 , K ) = 3( p f − 1) for trinomials when p > 2 + e .Fil: Avendaño, Martín. Texas A&M University; Estados UnidosFil: Krick, Teresa Elena Genoveva. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaElsevier2011-03info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/14919Avendaño, Martín; Krick, Teresa Elena Genoveva; Sharp bounds for the number of roots of univariate fewnomials; Elsevier; Journal Of Number Theory; 131; 7; 3-2011; 1209-12280022-314Xenginfo:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0022314X11000527info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jnt.2011.01.006info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:55:26Zoai:ri.conicet.gov.ar:11336/14919instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 09:55:27.132CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Sharp bounds for the number of roots of univariate fewnomials
title Sharp bounds for the number of roots of univariate fewnomials
spellingShingle Sharp bounds for the number of roots of univariate fewnomials
Avendaño, Martín
Lacunary Polynomials
Root Counting
Local Fields
Generealized Vandermonde Determinants
title_short Sharp bounds for the number of roots of univariate fewnomials
title_full Sharp bounds for the number of roots of univariate fewnomials
title_fullStr Sharp bounds for the number of roots of univariate fewnomials
title_full_unstemmed Sharp bounds for the number of roots of univariate fewnomials
title_sort Sharp bounds for the number of roots of univariate fewnomials
dc.creator.none.fl_str_mv Avendaño, Martín
Krick, Teresa Elena Genoveva
author Avendaño, Martín
author_facet Avendaño, Martín
Krick, Teresa Elena Genoveva
author_role author
author2 Krick, Teresa Elena Genoveva
author2_role author
dc.subject.none.fl_str_mv Lacunary Polynomials
Root Counting
Local Fields
Generealized Vandermonde Determinants
topic Lacunary Polynomials
Root Counting
Local Fields
Generealized Vandermonde Determinants
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Let K be a field and t ≥ 0. Denote by B m ( t, K ) the supremum of the number of roots in K ∗ , counted with multiplicities, that can have a non-zero polynomial in K [ x ] with at most t + 1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that B m ( t, L ) ≤ t 2 B m ( t, K ) for any local field L with a non-archimedean valuation v : L → R ∪{∞} such that v | Z 6 =0 ≡ 0 and residue field K , and that B m ( t, K ) ≤ ( t 2 − t +1)( p f − 1) for any finite extension K/ Q p with residual class degree f and ramification index e , assuming that p > t + e . For any finite extension K/ Q p , for p odd, we also show the lower bound B m ( t, K ) ≥ (2 t − 1)( p f − 1), which gives the sharp estimation B m (2 , K ) = 3( p f − 1) for trinomials when p > 2 + e .
Fil: Avendaño, Martín. Texas A&M University; Estados Unidos
Fil: Krick, Teresa Elena Genoveva. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
description Let K be a field and t ≥ 0. Denote by B m ( t, K ) the supremum of the number of roots in K ∗ , counted with multiplicities, that can have a non-zero polynomial in K [ x ] with at most t + 1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that B m ( t, L ) ≤ t 2 B m ( t, K ) for any local field L with a non-archimedean valuation v : L → R ∪{∞} such that v | Z 6 =0 ≡ 0 and residue field K , and that B m ( t, K ) ≤ ( t 2 − t +1)( p f − 1) for any finite extension K/ Q p with residual class degree f and ramification index e , assuming that p > t + e . For any finite extension K/ Q p , for p odd, we also show the lower bound B m ( t, K ) ≥ (2 t − 1)( p f − 1), which gives the sharp estimation B m (2 , K ) = 3( p f − 1) for trinomials when p > 2 + e .
publishDate 2011
dc.date.none.fl_str_mv 2011-03
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/11336/14919
Avendaño, Martín; Krick, Teresa Elena Genoveva; Sharp bounds for the number of roots of univariate fewnomials; Elsevier; Journal Of Number Theory; 131; 7; 3-2011; 1209-1228
0022-314X
url http://hdl.handle.net/11336/14919
identifier_str_mv Avendaño, Martín; Krick, Teresa Elena Genoveva; Sharp bounds for the number of roots of univariate fewnomials; Elsevier; Journal Of Number Theory; 131; 7; 3-2011; 1209-1228
0022-314X
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0022314X11000527
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jnt.2011.01.006
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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