On the intrinsic complexity of the arithmetic Nullstellensatz

Autores
Hägele, K.; Morais, J. E.; Pardo, L. M.; Sombra, Martín
Año de publicación
2000
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function πS(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.
Facultad de Ciencias Exactas
Materia
Matemática
Hilbert’s Nullstellensatz
estimaciones aritméticas
Nivel de accesibilidad
acceso abierto
Condiciones de uso
http://creativecommons.org/licenses/by-nc-sa/4.0/
Repositorio
SEDICI (UNLP)
Institución
Universidad Nacional de La Plata
OAI Identificador
oai:sedici.unlp.edu.ar:10915/83447

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spelling On the intrinsic complexity of the arithmetic NullstellensatzHägele, K.Morais, J. E.Pardo, L. M.Sombra, MartínMatemáticaHilbert’s Nullstellensatzestimaciones aritméticasWe show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function πS(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.Facultad de Ciencias Exactas2000info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionArticulohttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdf103-183http://sedici.unlp.edu.ar/handle/10915/83447enginfo:eu-repo/semantics/altIdentifier/issn/0022-4049info:eu-repo/semantics/altIdentifier/doi/10.1016/S0022-4049(98)00148-0info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-sa/4.0/Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)reponame:SEDICI (UNLP)instname:Universidad Nacional de La Platainstacron:UNLP2025-10-15T11:07:47Zoai:sedici.unlp.edu.ar:10915/83447Institucionalhttp://sedici.unlp.edu.ar/Universidad públicaNo correspondehttp://sedici.unlp.edu.ar/oai/snrdalira@sedici.unlp.edu.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:13292025-10-15 11:07:47.281SEDICI (UNLP) - Universidad Nacional de La Platafalse
dc.title.none.fl_str_mv On the intrinsic complexity of the arithmetic Nullstellensatz
title On the intrinsic complexity of the arithmetic Nullstellensatz
spellingShingle On the intrinsic complexity of the arithmetic Nullstellensatz
Hägele, K.
Matemática
Hilbert’s Nullstellensatz
estimaciones aritméticas
title_short On the intrinsic complexity of the arithmetic Nullstellensatz
title_full On the intrinsic complexity of the arithmetic Nullstellensatz
title_fullStr On the intrinsic complexity of the arithmetic Nullstellensatz
title_full_unstemmed On the intrinsic complexity of the arithmetic Nullstellensatz
title_sort On the intrinsic complexity of the arithmetic Nullstellensatz
dc.creator.none.fl_str_mv Hägele, K.
Morais, J. E.
Pardo, L. M.
Sombra, Martín
author Hägele, K.
author_facet Hägele, K.
Morais, J. E.
Pardo, L. M.
Sombra, Martín
author_role author
author2 Morais, J. E.
Pardo, L. M.
Sombra, Martín
author2_role author
author
author
dc.subject.none.fl_str_mv Matemática
Hilbert’s Nullstellensatz
estimaciones aritméticas
topic Matemática
Hilbert’s Nullstellensatz
estimaciones aritméticas
dc.description.none.fl_txt_mv We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function πS(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.
Facultad de Ciencias Exactas
description We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function πS(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.
publishDate 2000
dc.date.none.fl_str_mv 2000
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dc.language.none.fl_str_mv eng
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info:eu-repo/semantics/altIdentifier/doi/10.1016/S0022-4049(98)00148-0
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-sa/4.0/
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by-nc-sa/4.0/
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
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103-183
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