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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/14919
Ver los metadatos del registro completo
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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/ |
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application/pdf application/pdf application/pdf |
dc.publisher.none.fl_str_mv |
Elsevier |
publisher.none.fl_str_mv |
Elsevier |
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CONICET Digital (CONICET) |
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CONICET Digital (CONICET) |
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Consejo Nacional de Investigaciones Científicas y Técnicas |
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CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas |
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dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
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