Explicit estimates for polynomial systems defining irreducible smooth complete intersections

Autores
Von zur Gathen, Joachim; Matera, Guillermo
Año de publicación
2019
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
This paper deals with properties of the algebraic variety defined as the set of zeros of a “typical” sequence of polynomials. We consider various types of “nice” varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero “genericity” polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Here, the number of polynomials and their degrees are fixed. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace.
Fil: Von zur Gathen, Joachim. Universitat Bonn; Alemania
Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Materia
ABSOLUTE IRREDUCIBILITY
COMPLETE INTERSECTIONS
FINITE FIELDS
NONSINGULARITY
POLYNOMIAL SYSTEMS
Nivel de accesibilidad
acceso abierto
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:11336/149729

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network_name_str CONICET Digital (CONICET)
spelling Explicit estimates for polynomial systems defining irreducible smooth complete intersectionsVon zur Gathen, JoachimMatera, GuillermoABSOLUTE IRREDUCIBILITYCOMPLETE INTERSECTIONSFINITE FIELDSNONSINGULARITYPOLYNOMIAL SYSTEMShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1This paper deals with properties of the algebraic variety defined as the set of zeros of a “typical” sequence of polynomials. We consider various types of “nice” varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero “genericity” polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Here, the number of polynomials and their degrees are fixed. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace.Fil: Von zur Gathen, Joachim. Universitat Bonn; AlemaniaFil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaPolish Academy of Sciences. Institute of Mathematics2019-03-08info: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/11336/149729Von zur Gathen, Joachim; Matera, Guillermo; Explicit estimates for polynomial systems defining irreducible smooth complete intersections; Polish Academy of Sciences. Institute of Mathematics; Acta Arithmetica; 188; 3; 8-3-2019; 209-2400065-10361730-6264CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/188/3/112910/explicit-estimates-for-polynomial-systems-defining-irreducible-smooth-complete-intersectionsinfo:eu-repo/semantics/altIdentifier/doi/10.4064/aa8387-8-2018info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1512.05598info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:38:17Zoai:ri.conicet.gov.ar:11336/149729instacron: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 10:38:18.025CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Explicit estimates for polynomial systems defining irreducible smooth complete intersections
title Explicit estimates for polynomial systems defining irreducible smooth complete intersections
spellingShingle Explicit estimates for polynomial systems defining irreducible smooth complete intersections
Von zur Gathen, Joachim
ABSOLUTE IRREDUCIBILITY
COMPLETE INTERSECTIONS
FINITE FIELDS
NONSINGULARITY
POLYNOMIAL SYSTEMS
title_short Explicit estimates for polynomial systems defining irreducible smooth complete intersections
title_full Explicit estimates for polynomial systems defining irreducible smooth complete intersections
title_fullStr Explicit estimates for polynomial systems defining irreducible smooth complete intersections
title_full_unstemmed Explicit estimates for polynomial systems defining irreducible smooth complete intersections
title_sort Explicit estimates for polynomial systems defining irreducible smooth complete intersections
dc.creator.none.fl_str_mv Von zur Gathen, Joachim
Matera, Guillermo
author Von zur Gathen, Joachim
author_facet Von zur Gathen, Joachim
Matera, Guillermo
author_role author
author2 Matera, Guillermo
author2_role author
dc.subject.none.fl_str_mv ABSOLUTE IRREDUCIBILITY
COMPLETE INTERSECTIONS
FINITE FIELDS
NONSINGULARITY
POLYNOMIAL SYSTEMS
topic ABSOLUTE IRREDUCIBILITY
COMPLETE INTERSECTIONS
FINITE FIELDS
NONSINGULARITY
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 This paper deals with properties of the algebraic variety defined as the set of zeros of a “typical” sequence of polynomials. We consider various types of “nice” varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero “genericity” polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Here, the number of polynomials and their degrees are fixed. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace.
Fil: Von zur Gathen, Joachim. Universitat Bonn; Alemania
Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
description This paper deals with properties of the algebraic variety defined as the set of zeros of a “typical” sequence of polynomials. We consider various types of “nice” varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero “genericity” polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Here, the number of polynomials and their degrees are fixed. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace.
publishDate 2019
dc.date.none.fl_str_mv 2019-03-08
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/11336/149729
Von zur Gathen, Joachim; Matera, Guillermo; Explicit estimates for polynomial systems defining irreducible smooth complete intersections; Polish Academy of Sciences. Institute of Mathematics; Acta Arithmetica; 188; 3; 8-3-2019; 209-240
0065-1036
1730-6264
CONICET Digital
CONICET
url http://hdl.handle.net/11336/149729
identifier_str_mv Von zur Gathen, Joachim; Matera, Guillermo; Explicit estimates for polynomial systems defining irreducible smooth complete intersections; Polish Academy of Sciences. Institute of Mathematics; Acta Arithmetica; 188; 3; 8-3-2019; 209-240
0065-1036
1730-6264
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.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/188/3/112910/explicit-estimates-for-polynomial-systems-defining-irreducible-smooth-complete-intersections
info:eu-repo/semantics/altIdentifier/doi/10.4064/aa8387-8-2018
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1512.05598
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
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 Polish Academy of Sciences. Institute of Mathematics
publisher.none.fl_str_mv Polish Academy of Sciences. Institute of Mathematics
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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