Computing the Homology of Real Projective Sets

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise). Its cost depends on the condition of the input as well as on...

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Detalles Bibliográficos
Autores principales: Cucker, F., Krick, T., Shub, M.
Formato: JOUR
Materias:
Complexity
Condition
Exponential time
Homology groups
Real projective varieties
Computational methods
Mathematical techniques
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_16153375_v18_n4_p929_Cucker
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id todo:paper_16153375_v18_n4_p929_Cucker
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spelling todo:paper_16153375_v18_n4_p929_Cucker2023-10-03T16:28:15Z Computing the Homology of Real Projective Sets Cucker, F. Krick, T. Shub, M. Complexity Condition Exponential time Homology groups Real projective varieties Computational methods Complexity Condition Exponential time Homology groups Real projective varieties Mathematical techniques We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise). Its cost depends on the condition of the input as well as on its size and is singly exponential in the number of variables (the dimension of the ambient space) and polynomial in the condition and the degrees of the defining polynomials. In addition, we show that outside of an exceptional set of measure exponentially small in the size of the data, the algorithm takes exponential time. © 2017, SFoCM. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_16153375_v18_n4_p929_Cucker
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Complexity
Condition
Exponential time
Homology groups
Real projective varieties
Computational methods
Complexity
Condition
Exponential time
Homology groups
Real projective varieties
Mathematical techniques
spellingShingle Complexity
Condition
Exponential time
Homology groups
Real projective varieties
Computational methods
Complexity
Condition
Exponential time
Homology groups
Real projective varieties
Mathematical techniques
Cucker, F.
Krick, T.
Shub, M.
Computing the Homology of Real Projective Sets
topic_facet Complexity
Condition
Exponential time
Homology groups
Real projective varieties
Computational methods
Complexity
Condition
Exponential time
Homology groups
Real projective varieties
Mathematical techniques
description We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise). Its cost depends on the condition of the input as well as on its size and is singly exponential in the number of variables (the dimension of the ambient space) and polynomial in the condition and the degrees of the defining polynomials. In addition, we show that outside of an exceptional set of measure exponentially small in the size of the data, the algorithm takes exponential time. © 2017, SFoCM.
format JOUR
author Cucker, F.
Krick, T.
Shub, M.
author_facet Cucker, F.
Krick, T.
Shub, M.
author_sort Cucker, F.
title Computing the Homology of Real Projective Sets
title_short Computing the Homology of Real Projective Sets
title_full Computing the Homology of Real Projective Sets
title_fullStr Computing the Homology of Real Projective Sets
title_full_unstemmed Computing the Homology of Real Projective Sets
title_sort computing the homology of real projective sets
url http://hdl.handle.net/20.500.12110/paper_16153375_v18_n4_p929_Cucker
work_keys_str_mv AT cuckerf computingthehomologyofrealprojectivesets
AT krickt computingthehomologyofrealprojectivesets
AT shubm computingthehomologyofrealprojectivesets
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