Approximation on Nash sets with monomial singularities

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are...

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Detalles Bibliográficos
Autores: Baro González, Elías, Fernando Galván, José Francisco, Ruiz Sancho, Jesús María
Tipo de recurso: artículo
Fecha de publicación:2014
País:España
Institución:Universidad Complutense de Madrid (UCM)
Repositorio:Docta Complutense
Idioma:inglés
OAI Identifier:oai:docta.ucm.es:20.500.14352/33838
Acceso en línea:https://hdl.handle.net/20.500.14352/33838
Access Level:acceso abierto
Palabra clave:51
Semialgebraic
Nash
Approximation
Extension
Monomial singularity
Manifold with corners
Matemáticas (Matemáticas)
12 Matemáticas
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spelling Approximation on Nash sets with monomial singularitiesBaro González, ElíasFernando Galván, José FranciscoRuiz Sancho, Jesús María51SemialgebraicNashApproximationExtensionMonomial singularityManifold with cornersMatemáticas (Matemáticas)12 MatemáticasThis paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first "finiteness" and "weak normality" for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m^2, are also Nash diffeomorphic.ElsevierUniversidad Complutense de Madrid20142014-09-0120142014-09-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/33838reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/338382026-06-02T12:44:21Z
dc.title.none.fl_str_mv Approximation on Nash sets with monomial singularities
title Approximation on Nash sets with monomial singularities
spellingShingle Approximation on Nash sets with monomial singularities
Baro González, Elías
51
Semialgebraic
Nash
Approximation
Extension
Monomial singularity
Manifold with corners
Matemáticas (Matemáticas)
12 Matemáticas
title_short Approximation on Nash sets with monomial singularities
title_full Approximation on Nash sets with monomial singularities
title_fullStr Approximation on Nash sets with monomial singularities
title_full_unstemmed Approximation on Nash sets with monomial singularities
title_sort Approximation on Nash sets with monomial singularities
dc.creator.none.fl_str_mv Baro González, Elías
Fernando Galván, José Francisco
Ruiz Sancho, Jesús María
author Baro González, Elías
author_facet Baro González, Elías
Fernando Galván, José Francisco
Ruiz Sancho, Jesús María
author_role author
author2 Fernando Galván, José Francisco
Ruiz Sancho, Jesús María
author2_role author
author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 51
Semialgebraic
Nash
Approximation
Extension
Monomial singularity
Manifold with corners
Matemáticas (Matemáticas)
12 Matemáticas
topic 51
Semialgebraic
Nash
Approximation
Extension
Monomial singularity
Manifold with corners
Matemáticas (Matemáticas)
12 Matemáticas
description This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first "finiteness" and "weak normality" for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m^2, are also Nash diffeomorphic.
publishDate 2014
dc.date.none.fl_str_mv 2014
2014-09-01
2014
2014-09-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/33838
url https://hdl.handle.net/20.500.14352/33838
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300719

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