On the Multiplicity of Isolated Roots of Sparse Polynomial Systems

Autores
Herrero, Maria Isabel; Jeronimo, Gabriela Tali; Sabia, Juan Vicente Rafael
Año de publicación
2019
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these formulas to solve the problem in the general case, by showing that the multiplicity of an arbitrary affine isolated zero of a generic system with given supports equals the multiplicity of the origin as a common zero of a generic system with an associated family of supports. The formulas obtained are in the spirit of the classical Bernstein’s theorem, in the sense that they depend on the combinatorial structure of the system, namely, geometric numerical invariants associated to the supports, such as mixed volumes of convex sets and, alternatively, mixed integrals of convex functions.
Fil: Herrero, Maria Isabel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Jeronimo, Gabriela Tali. 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". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Sabia, Juan Vicente Rafael. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Ciclo Básico Común; Argentina
Materia
MIXED VOLUMES AND MIXED INTEGRALS
MULTIPLICITY OF ZEROS
NEWTON POLYTOPES
SPARSE POLYNOMIAL SYSTEMS
Nivel de accesibilidad
acceso embargado
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/89066
Acceder
id CONICETDig_059a02f9c8c3eff1cee23dee787d0663
oai_identifier_str oai:ri.conicet.gov.ar:11336/89066
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling On the Multiplicity of Isolated Roots of Sparse Polynomial SystemsHerrero, Maria IsabelJeronimo, Gabriela TaliSabia, Juan Vicente RafaelMIXED VOLUMES AND MIXED INTEGRALSMULTIPLICITY OF ZEROSNEWTON POLYTOPESSPARSE POLYNOMIAL SYSTEMShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these formulas to solve the problem in the general case, by showing that the multiplicity of an arbitrary affine isolated zero of a generic system with given supports equals the multiplicity of the origin as a common zero of a generic system with an associated family of supports. The formulas obtained are in the spirit of the classical Bernstein’s theorem, in the sense that they depend on the combinatorial structure of the system, namely, geometric numerical invariants associated to the supports, such as mixed volumes of convex sets and, alternatively, mixed integrals of convex functions.Fil: Herrero, Maria Isabel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Jeronimo, Gabriela Tali. 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". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Sabia, Juan Vicente Rafael. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Ciclo Básico Común; ArgentinaSpringer2019-12info:eu-repo/date/embargoEnd/2020-07-01info: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/11336/89066Herrero, Maria Isabel; Jeronimo, Gabriela Tali; Sabia, Juan Vicente Rafael; On the Multiplicity of Isolated Roots of Sparse Polynomial Systems; Springer; Discrete And Computational Geometry; 62; 4; 12-2019; 788-8120179-5376CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s00454-018-0025-xinfo:eu-repo/semantics/altIdentifier/doi/10.1007/s00454-018-0025-xinfo:eu-repo/semantics/embargoedAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:36:34Zoai:ri.conicet.gov.ar:11336/89066instacron: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 10:36:34.687CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
title On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
spellingShingle On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
Herrero, Maria Isabel
MIXED VOLUMES AND MIXED INTEGRALS
MULTIPLICITY OF ZEROS
NEWTON POLYTOPES
SPARSE POLYNOMIAL SYSTEMS
title_short On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
title_full On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
title_fullStr On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
title_full_unstemmed On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
title_sort On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
dc.creator.none.fl_str_mv Herrero, Maria Isabel
Jeronimo, Gabriela Tali
Sabia, Juan Vicente Rafael
author Herrero, Maria Isabel
author_facet Herrero, Maria Isabel
Jeronimo, Gabriela Tali
Sabia, Juan Vicente Rafael
author_role author
author2 Jeronimo, Gabriela Tali
Sabia, Juan Vicente Rafael
author2_role author
author
dc.subject.none.fl_str_mv MIXED VOLUMES AND MIXED INTEGRALS
MULTIPLICITY OF ZEROS
NEWTON POLYTOPES
SPARSE POLYNOMIAL SYSTEMS
topic MIXED VOLUMES AND MIXED INTEGRALS
MULTIPLICITY OF ZEROS
NEWTON POLYTOPES
SPARSE POLYNOMIAL SYSTEMS
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these formulas to solve the problem in the general case, by showing that the multiplicity of an arbitrary affine isolated zero of a generic system with given supports equals the multiplicity of the origin as a common zero of a generic system with an associated family of supports. The formulas obtained are in the spirit of the classical Bernstein’s theorem, in the sense that they depend on the combinatorial structure of the system, namely, geometric numerical invariants associated to the supports, such as mixed volumes of convex sets and, alternatively, mixed integrals of convex functions.
Fil: Herrero, Maria Isabel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Jeronimo, Gabriela Tali. 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". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Sabia, Juan Vicente Rafael. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Ciclo Básico Común; Argentina
description We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these formulas to solve the problem in the general case, by showing that the multiplicity of an arbitrary affine isolated zero of a generic system with given supports equals the multiplicity of the origin as a common zero of a generic system with an associated family of supports. The formulas obtained are in the spirit of the classical Bernstein’s theorem, in the sense that they depend on the combinatorial structure of the system, namely, geometric numerical invariants associated to the supports, such as mixed volumes of convex sets and, alternatively, mixed integrals of convex functions.
publishDate 2019
dc.date.none.fl_str_mv 2019-12
info:eu-repo/date/embargoEnd/2020-07-01
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/11336/89066
Herrero, Maria Isabel; Jeronimo, Gabriela Tali; Sabia, Juan Vicente Rafael; On the Multiplicity of Isolated Roots of Sparse Polynomial Systems; Springer; Discrete And Computational Geometry; 62; 4; 12-2019; 788-812
0179-5376
CONICET Digital
CONICET
url http://hdl.handle.net/11336/89066
identifier_str_mv Herrero, Maria Isabel; Jeronimo, Gabriela Tali; Sabia, Juan Vicente Rafael; On the Multiplicity of Isolated Roots of Sparse Polynomial Systems; Springer; Discrete And Computational Geometry; 62; 4; 12-2019; 788-812
0179-5376
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://link.springer.com/article/10.1007/s00454-018-0025-x
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00454-018-0025-x
dc.rights.none.fl_str_mv info:eu-repo/semantics/embargoedAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv embargoedAccess
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 Springer
publisher.none.fl_str_mv Springer
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
_version_ 1844614386374672384
score 13.070432

Publicaciones similares