On the value set of small families of polynomials over a finite field, I

Autores
Cesaratto, Eda; Matera, Guillermo; Pérez, Mariana; Privitelli, Melina Lorena
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients ad -1, ad-s are fixed. Our estimate holds without restrictions on the characteristic of Fq and asserts that V(d, s, a) = μdq+O(1), where V(d, s, a) is such an average cardinality, μd:=∑r=1d(-1)r-1/r! and a : = (a d-1,..., ad-s). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of Fq-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over Fq. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of Fq-rational points is established.
Fil: Cesaratto, Eda. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Matera, Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Pérez, Mariana. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Privitelli, Melina Lorena. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina
Materia
Average Value Set
Finite Fields
Rational Points
Singular Complete Intersections
Symmetric Polynomials
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:11336/54771
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/54771
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling On the value set of small families of polynomials over a finite field, ICesaratto, EdaMatera, GuillermoPérez, MarianaPrivitelli, Melina LorenaAverage Value SetFinite FieldsRational PointsSingular Complete IntersectionsSymmetric Polynomialshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients ad -1, ad-s are fixed. Our estimate holds without restrictions on the characteristic of Fq and asserts that V(d, s, a) = μdq+O(1), where V(d, s, a) is such an average cardinality, μd:=∑r=1d(-1)r-1/r! and a : = (a d-1,..., ad-s). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of Fq-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over Fq. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of Fq-rational points is established.Fil: Cesaratto, Eda. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Matera, Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Pérez, Mariana. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Privitelli, Melina Lorena. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaAcademic Press Inc Elsevier Science2014-05info: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/11336/54771Cesaratto, Eda; Matera, Guillermo; Pérez, Mariana; Privitelli, Melina Lorena; On the value set of small families of polynomials over a finite field, I; Academic Press Inc Elsevier Science; Journal of Combinatorial Theory Series A; 124; 1; 5-2014; 203-2270097-3165CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.jcta.2014.01.009info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S009731651400020Xinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-22T11:21:54Zoai:ri.conicet.gov.ar:11336/54771instacron: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:34982025-10-22 11:21:54.961CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On the value set of small families of polynomials over a finite field, I
title On the value set of small families of polynomials over a finite field, I
spellingShingle On the value set of small families of polynomials over a finite field, I
Cesaratto, Eda
Average Value Set
Finite Fields
Rational Points
Singular Complete Intersections
Symmetric Polynomials
title_short On the value set of small families of polynomials over a finite field, I
title_full On the value set of small families of polynomials over a finite field, I
title_fullStr On the value set of small families of polynomials over a finite field, I
title_full_unstemmed On the value set of small families of polynomials over a finite field, I
title_sort On the value set of small families of polynomials over a finite field, I
dc.creator.none.fl_str_mv Cesaratto, Eda
Matera, Guillermo
Pérez, Mariana
Privitelli, Melina Lorena
author Cesaratto, Eda
author_facet Cesaratto, Eda
Matera, Guillermo
Pérez, Mariana
Privitelli, Melina Lorena
author_role author
author2 Matera, Guillermo
Pérez, Mariana
Privitelli, Melina Lorena
author2_role author
author
author
dc.subject.none.fl_str_mv Average Value Set
Finite Fields
Rational Points
Singular Complete Intersections
Symmetric Polynomials
topic Average Value Set
Finite Fields
Rational Points
Singular Complete Intersections
Symmetric Polynomials
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 obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients ad -1, ad-s are fixed. Our estimate holds without restrictions on the characteristic of Fq and asserts that V(d, s, a) = μdq+O(1), where V(d, s, a) is such an average cardinality, μd:=∑r=1d(-1)r-1/r! and a : = (a d-1,..., ad-s). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of Fq-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over Fq. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of Fq-rational points is established.
Fil: Cesaratto, Eda. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Matera, Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Pérez, Mariana. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Privitelli, Melina Lorena. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina
description We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients ad -1, ad-s are fixed. Our estimate holds without restrictions on the characteristic of Fq and asserts that V(d, s, a) = μdq+O(1), where V(d, s, a) is such an average cardinality, μd:=∑r=1d(-1)r-1/r! and a : = (a d-1,..., ad-s). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of Fq-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over Fq. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of Fq-rational points is established.
publishDate 2014
dc.date.none.fl_str_mv 2014-05
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/11336/54771
Cesaratto, Eda; Matera, Guillermo; Pérez, Mariana; Privitelli, Melina Lorena; On the value set of small families of polynomials over a finite field, I; Academic Press Inc Elsevier Science; Journal of Combinatorial Theory Series A; 124; 1; 5-2014; 203-227
0097-3165
CONICET Digital
CONICET
url http://hdl.handle.net/11336/54771
identifier_str_mv Cesaratto, Eda; Matera, Guillermo; Pérez, Mariana; Privitelli, Melina Lorena; On the value set of small families of polynomials over a finite field, I; Academic Press Inc Elsevier Science; Journal of Combinatorial Theory Series A; 124; 1; 5-2014; 203-227
0097-3165
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jcta.2014.01.009
info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S009731651400020X
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
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score 12.982451

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