On the bit complexity of polynomial system solving

Autores
Gimenez, Nardo Ariel; Matera, Guillermo
Año de publicación
2019
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We describe and analyze a randomized algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence outside a given hypersurface. We show that its bit complexity is roughly quadratic in the Bézout number of the system and linear in its bit size. The algorithm solves the input system modulo a prime number p and applies p-adic lifting. For this purpose, we establish a number of results on the bit length of a “lucky” prime p, namely one for which the reduction of the input system modulo p preserves certain fundamental geometric and algebraic properties of the original system. These results rely on the analysis of Chow forms associated to the set of solutions of the input system and effective arithmetic Nullstellensätze.
Fil: Gimenez, Nardo Ariel. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Materia
BIT COMPLEXITY
CHOW FORM
LIFTING FIBERS
LUCKY PRIMES
POLYNOMIAL SYSTEM SOLVING OVER Q
REDUCED REGULAR SEQUENCE
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:11336/149498
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/149498
network_acronym_str CONICETDig
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network_name_str CONICET Digital (CONICET)
spelling On the bit complexity of polynomial system solvingGimenez, Nardo ArielMatera, GuillermoBIT COMPLEXITYCHOW FORMLIFTING FIBERSLUCKY PRIMESPOLYNOMIAL SYSTEM SOLVING OVER QREDUCED REGULAR SEQUENCEhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1https://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We describe and analyze a randomized algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence outside a given hypersurface. We show that its bit complexity is roughly quadratic in the Bézout number of the system and linear in its bit size. The algorithm solves the input system modulo a prime number p and applies p-adic lifting. For this purpose, we establish a number of results on the bit length of a “lucky” prime p, namely one for which the reduction of the input system modulo p preserves certain fundamental geometric and algebraic properties of the original system. These results rely on the analysis of Chow forms associated to the set of solutions of the input system and effective arithmetic Nullstellensätze.Fil: Gimenez, Nardo Ariel. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaAcademic Press Inc Elsevier Science2019-04info: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/11336/149498Gimenez, Nardo Ariel; Matera, Guillermo; On the bit complexity of polynomial system solving; Academic Press Inc Elsevier Science; Journal Of Complexity; 51; 4-2019; 20-670885-064X1090-2708CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0885064X18300761info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jco.2018.09.005info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T10:09:42Zoai:ri.conicet.gov.ar:11336/149498instacron: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:34982025-09-03 10:09:43.027CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On the bit complexity of polynomial system solving
title On the bit complexity of polynomial system solving
spellingShingle On the bit complexity of polynomial system solving
Gimenez, Nardo Ariel
BIT COMPLEXITY
CHOW FORM
LIFTING FIBERS
LUCKY PRIMES
POLYNOMIAL SYSTEM SOLVING OVER Q
REDUCED REGULAR SEQUENCE
title_short On the bit complexity of polynomial system solving
title_full On the bit complexity of polynomial system solving
title_fullStr On the bit complexity of polynomial system solving
title_full_unstemmed On the bit complexity of polynomial system solving
title_sort On the bit complexity of polynomial system solving
dc.creator.none.fl_str_mv Gimenez, Nardo Ariel
Matera, Guillermo
author Gimenez, Nardo Ariel
author_facet Gimenez, Nardo Ariel
Matera, Guillermo
author_role author
author2 Matera, Guillermo
author2_role author
dc.subject.none.fl_str_mv BIT COMPLEXITY
CHOW FORM
LIFTING FIBERS
LUCKY PRIMES
POLYNOMIAL SYSTEM SOLVING OVER Q
REDUCED REGULAR SEQUENCE
topic BIT COMPLEXITY
CHOW FORM
LIFTING FIBERS
LUCKY PRIMES
POLYNOMIAL SYSTEM SOLVING OVER Q
REDUCED REGULAR SEQUENCE
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We describe and analyze a randomized algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence outside a given hypersurface. We show that its bit complexity is roughly quadratic in the Bézout number of the system and linear in its bit size. The algorithm solves the input system modulo a prime number p and applies p-adic lifting. For this purpose, we establish a number of results on the bit length of a “lucky” prime p, namely one for which the reduction of the input system modulo p preserves certain fundamental geometric and algebraic properties of the original system. These results rely on the analysis of Chow forms associated to the set of solutions of the input system and effective arithmetic Nullstellensätze.
Fil: Gimenez, Nardo Ariel. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
description We describe and analyze a randomized algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence outside a given hypersurface. We show that its bit complexity is roughly quadratic in the Bézout number of the system and linear in its bit size. The algorithm solves the input system modulo a prime number p and applies p-adic lifting. For this purpose, we establish a number of results on the bit length of a “lucky” prime p, namely one for which the reduction of the input system modulo p preserves certain fundamental geometric and algebraic properties of the original system. These results rely on the analysis of Chow forms associated to the set of solutions of the input system and effective arithmetic Nullstellensätze.
publishDate 2019
dc.date.none.fl_str_mv 2019-04
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/11336/149498
Gimenez, Nardo Ariel; Matera, Guillermo; On the bit complexity of polynomial system solving; Academic Press Inc Elsevier Science; Journal Of Complexity; 51; 4-2019; 20-67
0885-064X
1090-2708
CONICET Digital
CONICET
url http://hdl.handle.net/11336/149498
identifier_str_mv Gimenez, Nardo Ariel; Matera, Guillermo; On the bit complexity of polynomial system solving; Academic Press Inc Elsevier Science; Journal Of Complexity; 51; 4-2019; 20-67
0885-064X
1090-2708
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.sciencedirect.com/science/article/pii/S0885064X18300761
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jco.2018.09.005
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
application/pdf
dc.publisher.none.fl_str_mv Academic Press Inc Elsevier Science
publisher.none.fl_str_mv Academic Press Inc Elsevier Science
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_ 1842270092168003584
score 13.13397

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