Identification of proofs via syzygies
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
DOI: | 10.1098/rsta.2018.0275 |
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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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:RCAAP2024-05-22T17:40:14Zoai:run.unl.pt:10362/75099Portal AgregadorONGhttps://www.rcaap.pt/oai/openairemluisa.alvim@gmail.comopendoar:71602024-05-22T17:40:14Repositório 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 Identification of proofs via syzygies Malheiro, António Hilbert’s 24th problem Identification of proofs Syzygies Mathematics(all) Engineering(all) Physics and Astronomy(all) 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 Identification of proofs via syzygies |
title_full_unstemmed |
Identification of proofs via syzygies Identification of proofs via syzygies |
title_sort |
Identification of proofs via syzygies |
author |
Malheiro, António |
author_facet |
Malheiro, António Malheiro, António Reis, José Francisco 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 |
dc.format.none.fl_str_mv |
11 application/pdf |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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 |
repository.mail.fl_str_mv |
mluisa.alvim@gmail.com |
_version_ |
1822183191452057600 |
dc.identifier.doi.none.fl_str_mv |
10.1098/rsta.2018.0275 |