Identification of proofs via syzygies

Detalhes bibliográficos
Autor(a) principal: Malheiro, António
Data de Publicação: 2019
Outros Autores: Reis, José Francisco
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: https://doi.org/10.1098/rsta.2018.0275
Resumo: For both authors, this work was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matematica e Aplicacoes) and the project PTDC/MHC-FIL/2583/2014. The first author was also funded by the FCT project PTDC/MAT-PUR/31174/2017.
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spelling Identification of proofs via syzygiesHilbert’s 24th problemIdentification of proofsSyzygiesMathematics(all)Engineering(all)Physics and Astronomy(all)For both authors, this work was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matematica e Aplicacoes) and the project PTDC/MHC-FIL/2583/2014. The first author was also funded by the FCT project PTDC/MAT-PUR/31174/2017.In 1900, Hilbert gave a lecture at the International Congress of Mathematicians in Paris, for which he prepared 23 problems that mathematicians should solve during the twentieth century. It was found that there was a note on a 24th problem focusing on the problem of simplicity of proofs. One of the lines of research that was generated from this problem was the identification of proofs. In this article, we present a possible method for exploring the identification of proofs based on the membership problem original from the theory of polynomial rings. To show this, we start by giving a complete worked-out example of a membership problem, that is the problem of checking if a given polynomial belongs to an ideal generated by finitely many polynomials. This problem can be solved by considering Gröbner bases and the corresponding reductions. Each reduction is a simplification of the polynomial and it corresponds to a rewriting step. In proving that a polynomial is a member of an ideal, a rewriting process is used, and many different such processes can be considered. To better illustrate this, we consider a graph where each rewriting step corresponds to an edge, and thus a path corresponds to a rewriting process. In this paper, we consider the identification of paths, within the context of the membership problem, to propose a criterion of identification of proofs.CMA - Centro de Matemática e AplicaçõesDM - Departamento de MatemáticaRUNMalheiro, AntónioReis, José Francisco2019-07-10T22:31:51Z2019-03-112019-03-11T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article11application/pdfhttps://doi.org/10.1098/rsta.2018.0275eng1364-503XPURE: 11608980http://www.scopus.com/inward/record.url?scp=85061315416&partnerID=8YFLogxKhttps://doi.org/10.1098/rsta.2018.0275info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-03-11T04:34:25Zoai:run.unl.pt:10362/75099Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T03:35:29.052285Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv Identification of proofs via syzygies
title Identification of proofs via syzygies
spellingShingle Identification of proofs via syzygies
Malheiro, António
Hilbert’s 24th problem
Identification of proofs
Syzygies
Mathematics(all)
Engineering(all)
Physics and Astronomy(all)
title_short Identification of proofs via syzygies
title_full Identification of proofs via syzygies
title_fullStr Identification of proofs via syzygies
title_full_unstemmed Identification of proofs via syzygies
title_sort Identification of proofs via syzygies
author Malheiro, António
author_facet Malheiro, António
Reis, José Francisco
author_role author
author2 Reis, José Francisco
author2_role author
dc.contributor.none.fl_str_mv CMA - Centro de Matemática e Aplicações
DM - Departamento de Matemática
RUN
dc.contributor.author.fl_str_mv Malheiro, António
Reis, José Francisco
dc.subject.por.fl_str_mv Hilbert’s 24th problem
Identification of proofs
Syzygies
Mathematics(all)
Engineering(all)
Physics and Astronomy(all)
topic Hilbert’s 24th problem
Identification of proofs
Syzygies
Mathematics(all)
Engineering(all)
Physics and Astronomy(all)
description For both authors, this work was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matematica e Aplicacoes) and the project PTDC/MHC-FIL/2583/2014. The first author was also funded by the FCT project PTDC/MAT-PUR/31174/2017.
publishDate 2019
dc.date.none.fl_str_mv 2019-07-10T22:31:51Z
2019-03-11
2019-03-11T00:00:00Z
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 https://doi.org/10.1098/rsta.2018.0275
url https://doi.org/10.1098/rsta.2018.0275
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 1364-503X
PURE: 11608980
http://www.scopus.com/inward/record.url?scp=85061315416&partnerID=8YFLogxK
https://doi.org/10.1098/rsta.2018.0275
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
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repository.name.fl_str_mv Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
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