Sums of squares on the hypercube
Autor(a) principal: | |
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Data de Publicação: | 2016 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10316/44180 https://doi.org/10.1007/s00209-016-1644-7 |
Resumo: | Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we are interested in rational function representations of small degree. We derive a general upper bound in terms of the Hilbert function of X, and we show that this upper bound is tight for the case of quadratic functions on the hypercube C={0,1}^n, a very well studied case in combinatorial optimization. Using the lower bounds for C we construct a family of globally nonnegative quartic polynomials, which are not sums of squares of rational functions of small degree. To our knowledge this is the first construction for Hilbert’s 17th problem of a family of polynomials of bounded degree which need increasing degrees in rational function representations as the number of variables n goes to infinity. We note that representation theory of the symmetric group S_n plays a crucial role in our proofs of the lower bounds. |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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7160 |
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Sums of squares on the hypercubeLet X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we are interested in rational function representations of small degree. We derive a general upper bound in terms of the Hilbert function of X, and we show that this upper bound is tight for the case of quadratic functions on the hypercube C={0,1}^n, a very well studied case in combinatorial optimization. Using the lower bounds for C we construct a family of globally nonnegative quartic polynomials, which are not sums of squares of rational functions of small degree. To our knowledge this is the first construction for Hilbert’s 17th problem of a family of polynomials of bounded degree which need increasing degrees in rational function representations as the number of variables n goes to infinity. We note that representation theory of the symmetric group S_n plays a crucial role in our proofs of the lower bounds.Springer2016info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://hdl.handle.net/10316/44180http://hdl.handle.net/10316/44180https://doi.org/10.1007/s00209-016-1644-7https://doi.org/10.1007/s00209-016-1644-7enghttps://doi.org/10.1007/s00209-016-1644-7Blekherman, GrigoriyGouveia, JoãoPfeiffer, Jamesinfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2021-06-29T10:02:50Zoai:estudogeral.uc.pt:10316/44180Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T20:53:31.600651Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Sums of squares on the hypercube |
title |
Sums of squares on the hypercube |
spellingShingle |
Sums of squares on the hypercube Blekherman, Grigoriy |
title_short |
Sums of squares on the hypercube |
title_full |
Sums of squares on the hypercube |
title_fullStr |
Sums of squares on the hypercube |
title_full_unstemmed |
Sums of squares on the hypercube |
title_sort |
Sums of squares on the hypercube |
author |
Blekherman, Grigoriy |
author_facet |
Blekherman, Grigoriy Gouveia, João Pfeiffer, James |
author_role |
author |
author2 |
Gouveia, João Pfeiffer, James |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Blekherman, Grigoriy Gouveia, João Pfeiffer, James |
description |
Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we are interested in rational function representations of small degree. We derive a general upper bound in terms of the Hilbert function of X, and we show that this upper bound is tight for the case of quadratic functions on the hypercube C={0,1}^n, a very well studied case in combinatorial optimization. Using the lower bounds for C we construct a family of globally nonnegative quartic polynomials, which are not sums of squares of rational functions of small degree. To our knowledge this is the first construction for Hilbert’s 17th problem of a family of polynomials of bounded degree which need increasing degrees in rational function representations as the number of variables n goes to infinity. We note that representation theory of the symmetric group S_n plays a crucial role in our proofs of the lower bounds. |
publishDate |
2016 |
dc.date.none.fl_str_mv |
2016 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10316/44180 http://hdl.handle.net/10316/44180 https://doi.org/10.1007/s00209-016-1644-7 https://doi.org/10.1007/s00209-016-1644-7 |
url |
http://hdl.handle.net/10316/44180 https://doi.org/10.1007/s00209-016-1644-7 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
https://doi.org/10.1007/s00209-016-1644-7 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Springer |
publisher.none.fl_str_mv |
Springer |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
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1799133821654794240 |