Topics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomials
Autor(a) principal: | |
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Data de Publicação: | 2023 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFMG |
Texto Completo: | http://hdl.handle.net/1843/51317 https://orcid.org/0000-0001-6809-6254 |
Resumo: | In this thesis we study some problems in the finite field theory that interesting for their applications in coding theory, cryptography, communications and related areas. Our first problem is to determine the number of rational points of a family of Artin-Schreier curves and of an Artin-Schreier hypersurface, as well as to determine conditions for these curves/hypersurface to be maximal or minimal with respect to the Hasse-Weil bound. In the sequence, we study a class of superelliptic curves and, under some conditions, we describe the number of rational points of these curves. The last topic of this work is about irreducible polynomials, where we determine conditions on n and q for which the irreducible factors over F_q of the binomial x^n-1 are binomials and trinomials. |
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Fabio Enrique Brochero Martínezhttp://lattes.cnpq.br/2118422761261421Daniel Nelson Panario RodriguezHerivelto Martins Borges FilhoLucas da Silva ReisLuciane Quoos ConteRicardo Alberto Podestáhttps://lattes.cnpq.br/9744143861766712Daniela Alves de Oliveira2023-03-29T14:22:37Z2023-03-29T14:22:37Z2023-03-10http://hdl.handle.net/1843/51317https://orcid.org/0000-0001-6809-6254In this thesis we study some problems in the finite field theory that interesting for their applications in coding theory, cryptography, communications and related areas. Our first problem is to determine the number of rational points of a family of Artin-Schreier curves and of an Artin-Schreier hypersurface, as well as to determine conditions for these curves/hypersurface to be maximal or minimal with respect to the Hasse-Weil bound. In the sequence, we study a class of superelliptic curves and, under some conditions, we describe the number of rational points of these curves. The last topic of this work is about irreducible polynomials, where we determine conditions on n and q for which the irreducible factors over F_q of the binomial x^n-1 are binomials and trinomials.Nesta tese estudamos alguns problemas da teoria de corpos finitos que são interessantes por suas aplicações em teoria de códigos, criptografia, comunicações e áreas relacionadas. Nosso primeiro problema é determinar o número de pontos racionais de uma família de curvas do tipo Artin-Schreier e de uma hipersuperfície de Artin-Schreier, assim como determinar condições para essas curvas/hipersuperfícies serem maximais ou minimais com respeito à cota de Hasse-Weil. Na sequência estudamos uma classe de curvas superelípticas e, sob algumas condições, descrevemos o número de pontos racionais dessas curvas. O último tópico deste trabalho é sobre polinômios irredutíveis, onde determinamos condições sobre n e q para os quais os fatores irredutíveis sobre F_q do binômio x^n-1 são binômios e trinômios.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorengUniversidade Federal de Minas GeraisPrograma de Pós-Graduação em MatemáticaUFMGBrasilICX - DEPARTAMENTO DE MATEMÁTICAMatemática – TesesCorpos finitos (Algebra) – TesesFormas Quadráticas – TesesSomas de Gauss – Teses.Finite FieldsQuadratic FormsArtin-Schreier's CurvesArtin-Schreier's HypersurfacesSuperelliptic CurvesHasse-Weil's BoundGauss SumsCirculant MatricesIrreducible PolynomialsTopics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomialsinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGORIGINALtese numerada_final.pdftese numerada_final.pdfapplication/pdf871368https://repositorio.ufmg.br/bitstream/1843/51317/1/tese%20numerada_final.pdf84729e036461890746315c9e8e753d78MD51LICENSElicense.txtlicense.txttext/plain; charset=utf-82118https://repositorio.ufmg.br/bitstream/1843/51317/2/license.txtcda590c95a0b51b4d15f60c9642ca272MD521843/513172023-03-29 11:22:38.003oai:repositorio.ufmg.br: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ório de PublicaçõesPUBhttps://repositorio.ufmg.br/oaiopendoar:2023-03-29T14:22:38Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false |
dc.title.pt_BR.fl_str_mv |
Topics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomials |
title |
Topics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomials |
spellingShingle |
Topics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomials Daniela Alves de Oliveira Finite Fields Quadratic Forms Artin-Schreier's Curves Artin-Schreier's Hypersurfaces Superelliptic Curves Hasse-Weil's Bound Gauss Sums Circulant Matrices Irreducible Polynomials Matemática – Teses Corpos finitos (Algebra) – Teses Formas Quadráticas – Teses Somas de Gauss – Teses. |
title_short |
Topics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomials |
title_full |
Topics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomials |
title_fullStr |
Topics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomials |
title_full_unstemmed |
Topics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomials |
title_sort |
Topics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomials |
author |
Daniela Alves de Oliveira |
author_facet |
Daniela Alves de Oliveira |
author_role |
author |
dc.contributor.advisor1.fl_str_mv |
Fabio Enrique Brochero Martínez |
dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/2118422761261421 |
dc.contributor.referee1.fl_str_mv |
Daniel Nelson Panario Rodriguez |
dc.contributor.referee2.fl_str_mv |
Herivelto Martins Borges Filho |
dc.contributor.referee3.fl_str_mv |
Lucas da Silva Reis |
dc.contributor.referee4.fl_str_mv |
Luciane Quoos Conte |
dc.contributor.referee5.fl_str_mv |
Ricardo Alberto Podestá |
dc.contributor.authorLattes.fl_str_mv |
https://lattes.cnpq.br/9744143861766712 |
dc.contributor.author.fl_str_mv |
Daniela Alves de Oliveira |
contributor_str_mv |
Fabio Enrique Brochero Martínez Daniel Nelson Panario Rodriguez Herivelto Martins Borges Filho Lucas da Silva Reis Luciane Quoos Conte Ricardo Alberto Podestá |
dc.subject.por.fl_str_mv |
Finite Fields Quadratic Forms Artin-Schreier's Curves Artin-Schreier's Hypersurfaces Superelliptic Curves Hasse-Weil's Bound Gauss Sums Circulant Matrices Irreducible Polynomials |
topic |
Finite Fields Quadratic Forms Artin-Schreier's Curves Artin-Schreier's Hypersurfaces Superelliptic Curves Hasse-Weil's Bound Gauss Sums Circulant Matrices Irreducible Polynomials Matemática – Teses Corpos finitos (Algebra) – Teses Formas Quadráticas – Teses Somas de Gauss – Teses. |
dc.subject.other.pt_BR.fl_str_mv |
Matemática – Teses Corpos finitos (Algebra) – Teses Formas Quadráticas – Teses Somas de Gauss – Teses. |
description |
In this thesis we study some problems in the finite field theory that interesting for their applications in coding theory, cryptography, communications and related areas. Our first problem is to determine the number of rational points of a family of Artin-Schreier curves and of an Artin-Schreier hypersurface, as well as to determine conditions for these curves/hypersurface to be maximal or minimal with respect to the Hasse-Weil bound. In the sequence, we study a class of superelliptic curves and, under some conditions, we describe the number of rational points of these curves. The last topic of this work is about irreducible polynomials, where we determine conditions on n and q for which the irreducible factors over F_q of the binomial x^n-1 are binomials and trinomials. |
publishDate |
2023 |
dc.date.accessioned.fl_str_mv |
2023-03-29T14:22:37Z |
dc.date.available.fl_str_mv |
2023-03-29T14:22:37Z |
dc.date.issued.fl_str_mv |
2023-03-10 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/1843/51317 |
dc.identifier.orcid.pt_BR.fl_str_mv |
https://orcid.org/0000-0001-6809-6254 |
url |
http://hdl.handle.net/1843/51317 https://orcid.org/0000-0001-6809-6254 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Minas Gerais |
dc.publisher.program.fl_str_mv |
Programa de Pós-Graduação em Matemática |
dc.publisher.initials.fl_str_mv |
UFMG |
dc.publisher.country.fl_str_mv |
Brasil |
dc.publisher.department.fl_str_mv |
ICX - DEPARTAMENTO DE MATEMÁTICA |
publisher.none.fl_str_mv |
Universidade Federal de Minas Gerais |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFMG instname:Universidade Federal de Minas Gerais (UFMG) instacron:UFMG |
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Universidade Federal de Minas Gerais (UFMG) |
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UFMG |
institution |
UFMG |
reponame_str |
Repositório Institucional da UFMG |
collection |
Repositório Institucional da UFMG |
bitstream.url.fl_str_mv |
https://repositorio.ufmg.br/bitstream/1843/51317/1/tese%20numerada_final.pdf https://repositorio.ufmg.br/bitstream/1843/51317/2/license.txt |
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Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG) |
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