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

Autores
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 ν(d, s, a) of the value set of any family of monic polynomials in double-struck Fq[T] of degree d for which s consecutive coefficients a = (ad-1, ..., ad-s) are fixed. Our estimate asserts that, ν(d, s, a) = μdq + O(q1/2), where μd := Σr=1d(-1)r-1/r!. We also prove that ν2(d, s, a) = μd2q2 + O(q3/2), where ν2(d, s, a) is the average second moment of the value set cardinalities for any family of monic polynomials of double-struck Fq[T] of degree d with s consecutive coefficients fixed as above. Finally, we show that ν2(d, 0) = μd2q2 + O(q), where ν2(d, 0) denotes the average second moment for all monic polynomials in double-struck Fq[T] of degree d with f(0) = 0. All our estimates hold for fields of characteristic p > 2 and provide explicit upper bounds for the O-constants in terms of d and s with "good" behavior. Our approach reduces the questions to estimating the number of double-struck Fq- rational points with pairwise distinct coordinates of a certain family of complete intersections defined over double-struck Fq. Critical to our results is the analysis of the singular locus of the varieties under consideration, which allows us obtain rather precise estimates on the corresponding number of double-struck Fq-rational points.
Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Pérez, Mariana. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Privitelli, Melina Lorena. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Materia
Average Second Moment
Average Value Set Cardinality
Finite Fields
Rational Points
Singular Complete Intersections
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:11336/55110
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/55110
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, IIMatera, GuillermoPérez, MarianaPrivitelli, Melina LorenaAverage Second MomentAverage Value Set CardinalityFinite FieldsRational PointsSingular Complete Intersectionshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We obtain an estimate on the average cardinality ν(d, s, a) of the value set of any family of monic polynomials in double-struck Fq[T] of degree d for which s consecutive coefficients a = (ad-1, ..., ad-s) are fixed. Our estimate asserts that, ν(d, s, a) = μdq + O(q1/2), where μd := Σr=1d(-1)r-1/r!. We also prove that ν2(d, s, a) = μd2q2 + O(q3/2), where ν2(d, s, a) is the average second moment of the value set cardinalities for any family of monic polynomials of double-struck Fq[T] of degree d with s consecutive coefficients fixed as above. Finally, we show that ν2(d, 0) = μd2q2 + O(q), where ν2(d, 0) denotes the average second moment for all monic polynomials in double-struck Fq[T] of degree d with f(0) = 0. All our estimates hold for fields of characteristic p > 2 and provide explicit upper bounds for the O-constants in terms of d and s with "good" behavior. Our approach reduces the questions to estimating the number of double-struck Fq- rational points with pairwise distinct coordinates of a certain family of complete intersections defined over double-struck Fq. Critical to our results is the analysis of the singular locus of the varieties under consideration, which allows us obtain rather precise estimates on the corresponding number of double-struck Fq-rational points.Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pérez, Mariana. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Privitelli, Melina Lorena. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaPolish Academy of Sciences. Institute of Mathematics2014-10info: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/11336/55110Matera, Guillermo; Pérez, Mariana; Privitelli, Melina Lorena; On the value set of small families of polynomials over a finite field, II; Polish Academy of Sciences. Institute of Mathematics; Acta Arithmetica; 165; 2; 10-2014; 141-1790065-10361730-6264CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/acta-arithmetica/all/165/2/83581/on-the-value-set-of-small-families-of-polynomials-over-a-finite-field-iiinfo:eu-repo/semantics/altIdentifier/doi/10.4064/aa165-2-3info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:33:14Zoai:ri.conicet.gov.ar:11336/55110instacron: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 09:33:14.407CONICET 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, II
title On the value set of small families of polynomials over a finite field, II
spellingShingle On the value set of small families of polynomials over a finite field, II
Matera, Guillermo
Average Second Moment
Average Value Set Cardinality
Finite Fields
Rational Points
Singular Complete Intersections
title_short On the value set of small families of polynomials over a finite field, II
title_full On the value set of small families of polynomials over a finite field, II
title_fullStr On the value set of small families of polynomials over a finite field, II
title_full_unstemmed On the value set of small families of polynomials over a finite field, II
title_sort On the value set of small families of polynomials over a finite field, II
dc.creator.none.fl_str_mv Matera, Guillermo
Pérez, Mariana
Privitelli, Melina Lorena
author Matera, Guillermo
author_facet Matera, Guillermo
Pérez, Mariana
Privitelli, Melina Lorena
author_role author
author2 Pérez, Mariana
Privitelli, Melina Lorena
author2_role author
author
dc.subject.none.fl_str_mv Average Second Moment
Average Value Set Cardinality
Finite Fields
Rational Points
Singular Complete Intersections
topic Average Second Moment
Average Value Set Cardinality
Finite Fields
Rational Points
Singular Complete Intersections
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 ν(d, s, a) of the value set of any family of monic polynomials in double-struck Fq[T] of degree d for which s consecutive coefficients a = (ad-1, ..., ad-s) are fixed. Our estimate asserts that, ν(d, s, a) = μdq + O(q1/2), where μd := Σr=1d(-1)r-1/r!. We also prove that ν2(d, s, a) = μd2q2 + O(q3/2), where ν2(d, s, a) is the average second moment of the value set cardinalities for any family of monic polynomials of double-struck Fq[T] of degree d with s consecutive coefficients fixed as above. Finally, we show that ν2(d, 0) = μd2q2 + O(q), where ν2(d, 0) denotes the average second moment for all monic polynomials in double-struck Fq[T] of degree d with f(0) = 0. All our estimates hold for fields of characteristic p > 2 and provide explicit upper bounds for the O-constants in terms of d and s with "good" behavior. Our approach reduces the questions to estimating the number of double-struck Fq- rational points with pairwise distinct coordinates of a certain family of complete intersections defined over double-struck Fq. Critical to our results is the analysis of the singular locus of the varieties under consideration, which allows us obtain rather precise estimates on the corresponding number of double-struck Fq-rational points.
Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Pérez, Mariana. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Privitelli, Melina Lorena. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
description We obtain an estimate on the average cardinality ν(d, s, a) of the value set of any family of monic polynomials in double-struck Fq[T] of degree d for which s consecutive coefficients a = (ad-1, ..., ad-s) are fixed. Our estimate asserts that, ν(d, s, a) = μdq + O(q1/2), where μd := Σr=1d(-1)r-1/r!. We also prove that ν2(d, s, a) = μd2q2 + O(q3/2), where ν2(d, s, a) is the average second moment of the value set cardinalities for any family of monic polynomials of double-struck Fq[T] of degree d with s consecutive coefficients fixed as above. Finally, we show that ν2(d, 0) = μd2q2 + O(q), where ν2(d, 0) denotes the average second moment for all monic polynomials in double-struck Fq[T] of degree d with f(0) = 0. All our estimates hold for fields of characteristic p > 2 and provide explicit upper bounds for the O-constants in terms of d and s with "good" behavior. Our approach reduces the questions to estimating the number of double-struck Fq- rational points with pairwise distinct coordinates of a certain family of complete intersections defined over double-struck Fq. Critical to our results is the analysis of the singular locus of the varieties under consideration, which allows us obtain rather precise estimates on the corresponding number of double-struck Fq-rational points.
publishDate 2014
dc.date.none.fl_str_mv 2014-10
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/11336/55110
Matera, Guillermo; Pérez, Mariana; Privitelli, Melina Lorena; On the value set of small families of polynomials over a finite field, II; Polish Academy of Sciences. Institute of Mathematics; Acta Arithmetica; 165; 2; 10-2014; 141-179
0065-1036
1730-6264
CONICET Digital
CONICET
url http://hdl.handle.net/11336/55110
identifier_str_mv Matera, Guillermo; Pérez, Mariana; Privitelli, Melina Lorena; On the value set of small families of polynomials over a finite field, II; Polish Academy of Sciences. Institute of Mathematics; Acta Arithmetica; 165; 2; 10-2014; 141-179
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/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/acta-arithmetica/all/165/2/83581/on-the-value-set-of-small-families-of-polynomials-over-a-finite-field-ii
info:eu-repo/semantics/altIdentifier/doi/10.4064/aa165-2-3
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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score 13.070432

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