The Canny–Emiris Conjecture for the Sparse Resultant

Autores
D'Andrea, Carlos; Jeronimo, Gabriela Tali; Sombra, Martín
Año de publicación
2022
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We present a product formula for the initial parts of the sparse resultant associated with an arbitrary family of supports, generalizing a previous result by Sturmfels. This allows to compute the homogeneities and degrees of this sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain an analogous product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated with a mixed subdivision of a polytope. Applying these results, we prove that under suitable hypothesis, the sparse resultant can be computed as the quotient of the determinant of such a square matrix by one of its principal minors. This generalizes the classical Macaulay formula for the homogeneous resultant and confirms a conjecture of Canny and Emiris.
Fil: D'Andrea, Carlos. Centre de Recerca Matemàtica; España. Universidad de Barcelona; España
Fil: Jeronimo, Gabriela Tali. 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: Sombra, Martín. Centre de Recerca Matemàtica; España. Institució Catalana de Recerca I Estudis Avançats; España. Universidad de Barcelona; España
Materia
INITIAL PART
MACAULAY FORMULA
MIXED SUBDIVISION
SPARSE RESULTANT
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/213781
Acceder
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network_name_str CONICET Digital (CONICET)
spelling The Canny–Emiris Conjecture for the Sparse ResultantD'Andrea, CarlosJeronimo, Gabriela TaliSombra, MartínINITIAL PARTMACAULAY FORMULAMIXED SUBDIVISIONSPARSE RESULTANThttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We present a product formula for the initial parts of the sparse resultant associated with an arbitrary family of supports, generalizing a previous result by Sturmfels. This allows to compute the homogeneities and degrees of this sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain an analogous product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated with a mixed subdivision of a polytope. Applying these results, we prove that under suitable hypothesis, the sparse resultant can be computed as the quotient of the determinant of such a square matrix by one of its principal minors. This generalizes the classical Macaulay formula for the homogeneous resultant and confirms a conjecture of Canny and Emiris.Fil: D'Andrea, Carlos. Centre de Recerca Matemàtica; España. Universidad de Barcelona; EspañaFil: Jeronimo, Gabriela Tali. 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: Sombra, Martín. Centre de Recerca Matemàtica; España. Institució Catalana de Recerca I Estudis Avançats; España. Universidad de Barcelona; EspañaSpringer2022-03info: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/11336/213781D'Andrea, Carlos; Jeronimo, Gabriela Tali; Sombra, Martín; The Canny–Emiris Conjecture for the Sparse Resultant; Springer; Foundations Of Computational Mathematics; 23; 3; 3-2022; 741-8011615-3375CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s10208-021-09547-3info:eu-repo/semantics/altIdentifier/doi/10.1007/s10208-021-09547-3info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-15T15:32:19Zoai:ri.conicet.gov.ar:11336/213781instacron: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:34982025-10-15 15:32:19.889CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv The Canny–Emiris Conjecture for the Sparse Resultant
title The Canny–Emiris Conjecture for the Sparse Resultant
spellingShingle The Canny–Emiris Conjecture for the Sparse Resultant
D'Andrea, Carlos
INITIAL PART
MACAULAY FORMULA
MIXED SUBDIVISION
SPARSE RESULTANT
title_short The Canny–Emiris Conjecture for the Sparse Resultant
title_full The Canny–Emiris Conjecture for the Sparse Resultant
title_fullStr The Canny–Emiris Conjecture for the Sparse Resultant
title_full_unstemmed The Canny–Emiris Conjecture for the Sparse Resultant
title_sort The Canny–Emiris Conjecture for the Sparse Resultant
dc.creator.none.fl_str_mv D'Andrea, Carlos
Jeronimo, Gabriela Tali
Sombra, Martín
author D'Andrea, Carlos
author_facet D'Andrea, Carlos
Jeronimo, Gabriela Tali
Sombra, Martín
author_role author
author2 Jeronimo, Gabriela Tali
Sombra, Martín
author2_role author
author
dc.subject.none.fl_str_mv INITIAL PART
MACAULAY FORMULA
MIXED SUBDIVISION
SPARSE RESULTANT
topic INITIAL PART
MACAULAY FORMULA
MIXED SUBDIVISION
SPARSE RESULTANT
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 present a product formula for the initial parts of the sparse resultant associated with an arbitrary family of supports, generalizing a previous result by Sturmfels. This allows to compute the homogeneities and degrees of this sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain an analogous product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated with a mixed subdivision of a polytope. Applying these results, we prove that under suitable hypothesis, the sparse resultant can be computed as the quotient of the determinant of such a square matrix by one of its principal minors. This generalizes the classical Macaulay formula for the homogeneous resultant and confirms a conjecture of Canny and Emiris.
Fil: D'Andrea, Carlos. Centre de Recerca Matemàtica; España. Universidad de Barcelona; España
Fil: Jeronimo, Gabriela Tali. 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: Sombra, Martín. Centre de Recerca Matemàtica; España. Institució Catalana de Recerca I Estudis Avançats; España. Universidad de Barcelona; España
description We present a product formula for the initial parts of the sparse resultant associated with an arbitrary family of supports, generalizing a previous result by Sturmfels. This allows to compute the homogeneities and degrees of this sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain an analogous product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated with a mixed subdivision of a polytope. Applying these results, we prove that under suitable hypothesis, the sparse resultant can be computed as the quotient of the determinant of such a square matrix by one of its principal minors. This generalizes the classical Macaulay formula for the homogeneous resultant and confirms a conjecture of Canny and Emiris.
publishDate 2022
dc.date.none.fl_str_mv 2022-03
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/11336/213781
D'Andrea, Carlos; Jeronimo, Gabriela Tali; Sombra, Martín; The Canny–Emiris Conjecture for the Sparse Resultant; Springer; Foundations Of Computational Mathematics; 23; 3; 3-2022; 741-801
1615-3375
CONICET Digital
CONICET
url http://hdl.handle.net/11336/213781
identifier_str_mv D'Andrea, Carlos; Jeronimo, Gabriela Tali; Sombra, Martín; The Canny–Emiris Conjecture for the Sparse Resultant; Springer; Foundations Of Computational Mathematics; 23; 3; 3-2022; 741-801
1615-3375
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/s10208-021-09547-3
info:eu-repo/semantics/altIdentifier/doi/10.1007/s10208-021-09547-3
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by/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
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score 13.22299

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