The distribution of factorization patterns on linear families of polynomials over a finite field

Autores
Cesaratto, Eda; Matera, Guillermo; Pérez, Mariana
Año de publicación
2017
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We estimate the number |Aλ| of elements on a linear family A of monic polynomials of Fq[T] of degree n having factorization pattern λ:=1λ12λ2nλn. We show that |Aλ| = T(λ)qn-m + O(qn-m-1/2), where T(λ) is the proportion of elements of the symmetric group of n elements with cycle pattern λ and m is the codimension of A. Furthermore, if the family A under consideration is “sparse”, then |Aλ|=T(λ)qn-m+O(qn-m-1). Our estimates hold for fields Fq of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the O-notation in terms of λ and A with “good” behavior. Our approach reduces the question to estimate the number of Fq-rational points of certain families of complete intersections defined over Fq. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of Fq-rational points are established.
Fil: Cesaratto, Eda. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Pérez, Mariana. Universidad Nacional de General Sarmiento; Argentina
Materia
Finite Fields
Polynomials
Factorization Patterns
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:11336/38592
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/38592
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling The distribution of factorization patterns on linear families of polynomials over a finite fieldCesaratto, EdaMatera, GuillermoPérez, MarianaFinite FieldsPolynomialsFactorization Patternshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We estimate the number |Aλ| of elements on a linear family A of monic polynomials of Fq[T] of degree n having factorization pattern λ:=1λ12λ2nλn. We show that |Aλ| = T(λ)qn-m + O(qn-m-1/2), where T(λ) is the proportion of elements of the symmetric group of n elements with cycle pattern λ and m is the codimension of A. Furthermore, if the family A under consideration is “sparse”, then |Aλ|=T(λ)qn-m+O(qn-m-1). Our estimates hold for fields Fq of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the O-notation in terms of λ and A with “good” behavior. Our approach reduces the question to estimate the number of Fq-rational points of certain families of complete intersections defined over Fq. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of Fq-rational points are established.Fil: Cesaratto, Eda. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Matera, Guillermo. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pérez, Mariana. Universidad Nacional de General Sarmiento; ArgentinaSpringer2017-10info: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/11336/38592Cesaratto, Eda; Matera, Guillermo; Pérez, Mariana; The distribution of factorization patterns on linear families of polynomials over a finite field; Springer; Combinatorica; 37; 5; 10-2017; 805-8360209-9683CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1007/s00493-015-3330-5info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00493-015-3330-5info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/pdf/1408.7014.pdfinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:23:45Zoai:ri.conicet.gov.ar:11336/38592instacron: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 10:23:46.119CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv The distribution of factorization patterns on linear families of polynomials over a finite field
title The distribution of factorization patterns on linear families of polynomials over a finite field
spellingShingle The distribution of factorization patterns on linear families of polynomials over a finite field
Cesaratto, Eda
Finite Fields
Polynomials
Factorization Patterns
title_short The distribution of factorization patterns on linear families of polynomials over a finite field
title_full The distribution of factorization patterns on linear families of polynomials over a finite field
title_fullStr The distribution of factorization patterns on linear families of polynomials over a finite field
title_full_unstemmed The distribution of factorization patterns on linear families of polynomials over a finite field
title_sort The distribution of factorization patterns on linear families of polynomials over a finite field
dc.creator.none.fl_str_mv Cesaratto, Eda
Matera, Guillermo
Pérez, Mariana
author Cesaratto, Eda
author_facet Cesaratto, Eda
Matera, Guillermo
Pérez, Mariana
author_role author
author2 Matera, Guillermo
Pérez, Mariana
author2_role author
author
dc.subject.none.fl_str_mv Finite Fields
Polynomials
Factorization Patterns
topic Finite Fields
Polynomials
Factorization Patterns
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 estimate the number |Aλ| of elements on a linear family A of monic polynomials of Fq[T] of degree n having factorization pattern λ:=1λ12λ2nλn. We show that |Aλ| = T(λ)qn-m + O(qn-m-1/2), where T(λ) is the proportion of elements of the symmetric group of n elements with cycle pattern λ and m is the codimension of A. Furthermore, if the family A under consideration is “sparse”, then |Aλ|=T(λ)qn-m+O(qn-m-1). Our estimates hold for fields Fq of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the O-notation in terms of λ and A with “good” behavior. Our approach reduces the question to estimate the number of Fq-rational points of certain families of complete intersections defined over Fq. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of Fq-rational points are established.
Fil: Cesaratto, Eda. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Pérez, Mariana. Universidad Nacional de General Sarmiento; Argentina
description We estimate the number |Aλ| of elements on a linear family A of monic polynomials of Fq[T] of degree n having factorization pattern λ:=1λ12λ2nλn. We show that |Aλ| = T(λ)qn-m + O(qn-m-1/2), where T(λ) is the proportion of elements of the symmetric group of n elements with cycle pattern λ and m is the codimension of A. Furthermore, if the family A under consideration is “sparse”, then |Aλ|=T(λ)qn-m+O(qn-m-1). Our estimates hold for fields Fq of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the O-notation in terms of λ and A with “good” behavior. Our approach reduces the question to estimate the number of Fq-rational points of certain families of complete intersections defined over Fq. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of Fq-rational points are established.
publishDate 2017
dc.date.none.fl_str_mv 2017-10
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/11336/38592
Cesaratto, Eda; Matera, Guillermo; Pérez, Mariana; The distribution of factorization patterns on linear families of polynomials over a finite field; Springer; Combinatorica; 37; 5; 10-2017; 805-836
0209-9683
CONICET Digital
CONICET
url http://hdl.handle.net/11336/38592
identifier_str_mv Cesaratto, Eda; Matera, Guillermo; Pérez, Mariana; The distribution of factorization patterns on linear families of polynomials over a finite field; Springer; Combinatorica; 37; 5; 10-2017; 805-836
0209-9683
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1007/s00493-015-3330-5
info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00493-015-3330-5
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/pdf/1408.7014.pdf
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 Springer
publisher.none.fl_str_mv Springer
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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