Tropicalization of facets of polytopes

Autores
Allamigeon, Xavier; Katz, Ricardo David
Año de publicación
2017
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu.
Fil: Allamigeon, Xavier. Ecole Polytechnique. Centre de Mathematiques Appliquees; Francia. Institut National de Recherche en Informatique et en Automatique; Francia. Centre National de la Recherche Scientifique; Francia
Fil: Katz, Ricardo David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentina
Materia
EXTERNAL REPRESENTATIONS
FACET-DEFINING HALF-SPACES
HAHN SERIES FIELD
POLYTOPES
PUISEUX SERIES FIELD
TROPICAL CONVEXITY
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:11336/53346
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/53346
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Tropicalization of facets of polytopesAllamigeon, XavierKatz, Ricardo DavidEXTERNAL REPRESENTATIONSFACET-DEFINING HALF-SPACESHAHN SERIES FIELDPOLYTOPESPUISEUX SERIES FIELDTROPICAL CONVEXITYhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu.Fil: Allamigeon, Xavier. Ecole Polytechnique. Centre de Mathematiques Appliquees; Francia. Institut National de Recherche en Informatique et en Automatique; Francia. Centre National de la Recherche Scientifique; FranciaFil: Katz, Ricardo David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; ArgentinaElsevier Science Inc2017-06info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/53346Allamigeon, Xavier; Katz, Ricardo David; Tropicalization of facets of polytopes; Elsevier Science Inc; Linear Algebra and its Applications; 523; 6-2017; 79-1010024-3795CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1408.6176info:eu-repo/semantics/altIdentifier/doi/10.1016/j.laa.2017.02.011info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0024379517300873info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:34:03Zoai:ri.conicet.gov.ar:11336/53346instacron: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 09:34:03.386CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Tropicalization of facets of polytopes
title Tropicalization of facets of polytopes
spellingShingle Tropicalization of facets of polytopes
Allamigeon, Xavier
EXTERNAL REPRESENTATIONS
FACET-DEFINING HALF-SPACES
HAHN SERIES FIELD
POLYTOPES
PUISEUX SERIES FIELD
TROPICAL CONVEXITY
title_short Tropicalization of facets of polytopes
title_full Tropicalization of facets of polytopes
title_fullStr Tropicalization of facets of polytopes
title_full_unstemmed Tropicalization of facets of polytopes
title_sort Tropicalization of facets of polytopes
dc.creator.none.fl_str_mv Allamigeon, Xavier
Katz, Ricardo David
author Allamigeon, Xavier
author_facet Allamigeon, Xavier
Katz, Ricardo David
author_role author
author2 Katz, Ricardo David
author2_role author
dc.subject.none.fl_str_mv EXTERNAL REPRESENTATIONS
FACET-DEFINING HALF-SPACES
HAHN SERIES FIELD
POLYTOPES
PUISEUX SERIES FIELD
TROPICAL CONVEXITY
topic EXTERNAL REPRESENTATIONS
FACET-DEFINING HALF-SPACES
HAHN SERIES FIELD
POLYTOPES
PUISEUX SERIES FIELD
TROPICAL CONVEXITY
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu.
Fil: Allamigeon, Xavier. Ecole Polytechnique. Centre de Mathematiques Appliquees; Francia. Institut National de Recherche en Informatique et en Automatique; Francia. Centre National de la Recherche Scientifique; Francia
Fil: Katz, Ricardo David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentina
description It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu.
publishDate 2017
dc.date.none.fl_str_mv 2017-06
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/11336/53346
Allamigeon, Xavier; Katz, Ricardo David; Tropicalization of facets of polytopes; Elsevier Science Inc; Linear Algebra and its Applications; 523; 6-2017; 79-101
0024-3795
CONICET Digital
CONICET
url http://hdl.handle.net/11336/53346
identifier_str_mv Allamigeon, Xavier; Katz, Ricardo David; Tropicalization of facets of polytopes; Elsevier Science Inc; Linear Algebra and its Applications; 523; 6-2017; 79-101
0024-3795
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://arxiv.org/abs/1408.6176
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.laa.2017.02.011
info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0024379517300873
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
dc.publisher.none.fl_str_mv Elsevier Science Inc
publisher.none.fl_str_mv Elsevier Science Inc
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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score 13.070432

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