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
- Institución
- Universidad Nacional de La Plata
- OAI Identificador
- oai:sedici.unlp.edu.ar:10915/83447
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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 |
dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Articulo 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://sedici.unlp.edu.ar/handle/10915/83447 |
url |
http://sedici.unlp.edu.ar/handle/10915/83447 |
dc.language.none.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
info:eu-repo/semantics/altIdentifier/issn/0022-4049 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) |
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openAccess |
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http://creativecommons.org/licenses/by-nc-sa/4.0/ Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) |
dc.format.none.fl_str_mv |
application/pdf 103-183 |
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