Lower bounds for norms of products of polynomials via Bombieri inequality

In this paper we give a different interpretation of Bombieri´s norm. This new point of view allows us to work on a problem posed by Beauzamy about the behavior of the sequence $S_n(P)=sup_{Q_n}, [PQ_n]_2$, where $P$ is a fixed $m-$homogeneous polynomial and $Q_n$ runs over the unit ball of the Hilbe...

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Detalles Bibliográficos
Autor: Pinasco, Damian
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2012
País:Argentina
Institución:Consejo Nacional de Investigaciones Científicas y Técnicas
Repositorio:CONICET Digital (CONICET)
Idioma:inglés
OAI Identifier:oai:ri.conicet.gov.ar:11336/196933
Acceso en línea:http://hdl.handle.net/11336/196933
Access Level:acceso abierto
Palabra clave:BOMBIERI'S INEQUALITY
BOMBIERI'S NORM
GAUSSIAN MEASURE
PLANK PROBLEM
POLYNOMIAL
PRODUCTS OF LINEAR FUNCTIONALS
UNIFORM NORMS INEQUALITIES
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
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spelling Lower bounds for norms of products of polynomials via Bombieri inequalityPinasco, DamianBOMBIERI'S INEQUALITYBOMBIERI'S NORMGAUSSIAN MEASUREPLANK PROBLEMPOLYNOMIALPRODUCTS OF LINEAR FUNCTIONALSUNIFORM NORMS INEQUALITIEShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this paper we give a different interpretation of Bombieri´s norm. This new point of view allows us to work on a problem posed by Beauzamy about the behavior of the sequence $S_n(P)=sup_{Q_n}, [PQ_n]_2$, where $P$ is a fixed $m-$homogeneous polynomial and $Q_n$ runs over the unit ball of the Hilbert space of $n-$homogeneous polynomials. We also study the factor problem for homogeneous  polynomials defined on $zC^N$ and we obtain sharp inequalities whenever the number of factors is no greater than $N$. In particular, we prove that for the product of homogeneous polynomials on infinite dimensional complex Hilbert spaces our inequality is sharp. Finally, we use these ideas to prove that any set ${z_k}_{k=1}^n$ of unit vectors in a complex Hilbert space for which $sup_{Vert z Vert=1} vert langle z, z_1 angle cdots langle z, z_n angle vert$ is minimum must be an orthonormal system.Fil: Pinasco, Damian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaAmerican Mathematical Society2012-08info: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/11336/196933Pinasco, Damian; Lower bounds for norms of products of polynomials via Bombieri inequality; American Mathematical Society; Transactions of the American Mathematical Society; 364; 8; 8-2012; 3393-40100002-9947CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.ams.org/journals/tran/2012-364-08/S0002-9947-2012-05403-1/S0002-9947-2012-05403-1.pdfinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2024-05-08T13:47:00Zoai:ri.conicet.gov.ar:11336/196933instacron: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:34982024-05-08 13:47:00.21CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Lower bounds for norms of products of polynomials via Bombieri inequality
title Lower bounds for norms of products of polynomials via Bombieri inequality
spellingShingle Lower bounds for norms of products of polynomials via Bombieri inequality
Pinasco, Damian
BOMBIERI'S INEQUALITY
BOMBIERI'S NORM
GAUSSIAN MEASURE
PLANK PROBLEM
POLYNOMIAL
PRODUCTS OF LINEAR FUNCTIONALS
UNIFORM NORMS INEQUALITIES
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
title_short Lower bounds for norms of products of polynomials via Bombieri inequality
title_full Lower bounds for norms of products of polynomials via Bombieri inequality
title_fullStr Lower bounds for norms of products of polynomials via Bombieri inequality
title_full_unstemmed Lower bounds for norms of products of polynomials via Bombieri inequality
title_sort Lower bounds for norms of products of polynomials via Bombieri inequality
dc.creator.none.fl_str_mv Pinasco, Damian
author Pinasco, Damian
author_facet Pinasco, Damian
author_role author
dc.subject.none.fl_str_mv BOMBIERI'S INEQUALITY
BOMBIERI'S NORM
GAUSSIAN MEASURE
PLANK PROBLEM
POLYNOMIAL
PRODUCTS OF LINEAR FUNCTIONALS
UNIFORM NORMS INEQUALITIES
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
topic BOMBIERI'S INEQUALITY
BOMBIERI'S NORM
GAUSSIAN MEASURE
PLANK PROBLEM
POLYNOMIAL
PRODUCTS OF LINEAR FUNCTIONALS
UNIFORM NORMS INEQUALITIES
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
description In this paper we give a different interpretation of Bombieri´s norm. This new point of view allows us to work on a problem posed by Beauzamy about the behavior of the sequence $S_n(P)=sup_{Q_n}, [PQ_n]_2$, where $P$ is a fixed $m-$homogeneous polynomial and $Q_n$ runs over the unit ball of the Hilbert space of $n-$homogeneous polynomials. We also study the factor problem for homogeneous  polynomials defined on $zC^N$ and we obtain sharp inequalities whenever the number of factors is no greater than $N$. In particular, we prove that for the product of homogeneous polynomials on infinite dimensional complex Hilbert spaces our inequality is sharp. Finally, we use these ideas to prove that any set ${z_k}_{k=1}^n$ of unit vectors in a complex Hilbert space for which $sup_{Vert z Vert=1} vert langle z, z_1 angle cdots langle z, z_n angle vert$ is minimum must be an orthonormal system.
publishDate 2012
dc.date.none.fl_str_mv 2012-08
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/11336/196933
Pinasco, Damian; Lower bounds for norms of products of polynomials via Bombieri inequality; American Mathematical Society; Transactions of the American Mathematical Society; 364; 8; 8-2012; 3393-4010
0002-9947
CONICET Digital
CONICET
url http://hdl.handle.net/11336/196933
identifier_str_mv Pinasco, Damian; Lower bounds for norms of products of polynomials via Bombieri inequality; American Mathematical Society; Transactions of the American Mathematical Society; 364; 8; 8-2012; 3393-4010
0002-9947
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.ams.org/journals/tran/2012-364-08/S0002-9947-2012-05403-1/S0002-9947-2012-05403-1.pdf
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv American Mathematical Society
publisher.none.fl_str_mv American Mathematical Society
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
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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score 15,811543

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