On well-rounded lattices and lower bounds for the minimum norm of ideal lattices

Bibliographic Details
Main Author: Alves, Carina [UNESP]
Publication Date: 2025
Other Authors: Strapasson, João E., Araujo, Robson R. de
Format: Article
Language: eng
Source: Repositório Institucional da UNESP
Download full: http://dx.doi.org/10.1007/s00013-024-02065-y
https://hdl.handle.net/11449/307139
Summary: In this paper, we study properties of well-rounded ideal lattices focusing on the lower bounds for their minimum norm. We present counterexamples showing that the stated bounds in a previous work do not hold for mixed number fields through the canonical embedding. However, we prove that ideal lattices obtained via the Minkowski embedding (instead of the canonical embedding) are well-rounded if and only if the number field is cyclotomic. Additionally, we derive new lower bounds for the minimum norm of ideal lattices under both the canonical and twisted embeddings. Our results not only refine existing theories but also open new possibilities for research on well-rounded ideal lattices in higher dimensions.
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spelling On well-rounded lattices and lower bounds for the minimum norm of ideal latticesCyclotomic fieldsIdeal latticesMinimum normNumber fieldsWell-rounded latticesIn this paper, we study properties of well-rounded ideal lattices focusing on the lower bounds for their minimum norm. We present counterexamples showing that the stated bounds in a previous work do not hold for mixed number fields through the canonical embedding. However, we prove that ideal lattices obtained via the Minkowski embedding (instead of the canonical embedding) are well-rounded if and only if the number field is cyclotomic. Additionally, we derive new lower bounds for the minimum norm of ideal lattices under both the canonical and twisted embeddings. Our results not only refine existing theories but also open new possibilities for research on well-rounded ideal lattices in higher dimensions.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Department of Mathematics São Paulo State University (UNESP), 1515, 24A Avenue, SPFaculty of Applied Sciences University of Campinas (Unicamp), 1300, Pedro Zaccaria Street, SPFederal Institute of São Paulo (IFSP), 239, Pastor José Dutra de Moraes Street, SPDepartment of Mathematics São Paulo State University (UNESP), 1515, 24A Avenue, SPFAPESP: 2020/09838-0FAPESP: 2022/12667-9FAPESP: 2024/03333-5CNPq: 405842/2023-6Universidade Estadual Paulista (UNESP)Universidade Estadual de Campinas (UNICAMP)Federal Institute of São Paulo (IFSP)Alves, Carina [UNESP]Strapasson, João E.Araujo, Robson R. de2025-04-29T20:08:33Z2025-02-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article121-130http://dx.doi.org/10.1007/s00013-024-02065-yArchiv der Mathematik, v. 124, n. 2, p. 121-130, 2025.1420-89380003-889Xhttps://hdl.handle.net/11449/30713910.1007/s00013-024-02065-y2-s2.0-85207963580Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengArchiv der Mathematikinfo:eu-repo/semantics/openAccess2025-04-30T14:00:18Zoai:repositorio.unesp.br:11449/307139Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestrepositoriounesp@unesp.bropendoar:29462025-04-30T14:00:18Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false
dc.title.none.fl_str_mv On well-rounded lattices and lower bounds for the minimum norm of ideal lattices
title On well-rounded lattices and lower bounds for the minimum norm of ideal lattices
spellingShingle On well-rounded lattices and lower bounds for the minimum norm of ideal lattices
Alves, Carina [UNESP]
Cyclotomic fields
Ideal lattices
Minimum norm
Number fields
Well-rounded lattices
title_short On well-rounded lattices and lower bounds for the minimum norm of ideal lattices
title_full On well-rounded lattices and lower bounds for the minimum norm of ideal lattices
title_fullStr On well-rounded lattices and lower bounds for the minimum norm of ideal lattices
title_full_unstemmed On well-rounded lattices and lower bounds for the minimum norm of ideal lattices
title_sort On well-rounded lattices and lower bounds for the minimum norm of ideal lattices
author Alves, Carina [UNESP]
author_facet Alves, Carina [UNESP]
Strapasson, João E.
Araujo, Robson R. de
author_role author
author2 Strapasson, João E.
Araujo, Robson R. de
author2_role author
author
dc.contributor.none.fl_str_mv Universidade Estadual Paulista (UNESP)
Universidade Estadual de Campinas (UNICAMP)
Federal Institute of São Paulo (IFSP)
dc.contributor.author.fl_str_mv Alves, Carina [UNESP]
Strapasson, João E.
Araujo, Robson R. de
dc.subject.por.fl_str_mv Cyclotomic fields
Ideal lattices
Minimum norm
Number fields
Well-rounded lattices
topic Cyclotomic fields
Ideal lattices
Minimum norm
Number fields
Well-rounded lattices
description In this paper, we study properties of well-rounded ideal lattices focusing on the lower bounds for their minimum norm. We present counterexamples showing that the stated bounds in a previous work do not hold for mixed number fields through the canonical embedding. However, we prove that ideal lattices obtained via the Minkowski embedding (instead of the canonical embedding) are well-rounded if and only if the number field is cyclotomic. Additionally, we derive new lower bounds for the minimum norm of ideal lattices under both the canonical and twisted embeddings. Our results not only refine existing theories but also open new possibilities for research on well-rounded ideal lattices in higher dimensions.
publishDate 2025
dc.date.none.fl_str_mv 2025-04-29T20:08:33Z
2025-02-01
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://dx.doi.org/10.1007/s00013-024-02065-y
Archiv der Mathematik, v. 124, n. 2, p. 121-130, 2025.
1420-8938
0003-889X
https://hdl.handle.net/11449/307139
10.1007/s00013-024-02065-y
2-s2.0-85207963580
url http://dx.doi.org/10.1007/s00013-024-02065-y
https://hdl.handle.net/11449/307139
identifier_str_mv Archiv der Mathematik, v. 124, n. 2, p. 121-130, 2025.
1420-8938
0003-889X
10.1007/s00013-024-02065-y
2-s2.0-85207963580
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Archiv der Mathematik
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 121-130
dc.source.none.fl_str_mv Scopus
reponame:Repositório Institucional da UNESP
instname:Universidade Estadual Paulista (UNESP)
instacron:UNESP
instname_str Universidade Estadual Paulista (UNESP)
instacron_str UNESP
institution UNESP
reponame_str Repositório Institucional da UNESP
collection Repositório Institucional da UNESP
repository.name.fl_str_mv Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)
repository.mail.fl_str_mv repositoriounesp@unesp.br
_version_ 1834482932093288448

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