Computing isolated roots of sparse polynomial systems in affine space

We present a symbolic probabilistic algorithm to compute the isolated roots in Cn of sparse polynomial equation systems. As some already known numerical algorithms solving this task, our procedure is based on polyhedral deformations and homotopies, but it amounts to solving a smaller number of squar...

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Autores principales: Herrero, María Isabel, Jeronimo, Gabriela Tali, Sabia, Juan Vicente Rafael
Publicado: 2010
Materias:
Algorithms
Complexity
Sparse polynomial systems
Affine space
Combinatorial structures
Finite set
Geometric resolution
Homotopies
Numerical algorithms
Pre-processing
Probabilistic algorithm
Sparse polynomials
System supports
Systems of equations
Topology
Polynomials
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03043975_v411_n44-46_p3894_Herrero
http://hdl.handle.net/20.500.12110/paper_03043975_v411_n44-46_p3894_Herrero
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_03043975_v411_n44-46_p3894_Herrero
record_format dspace
spelling paper:paper_03043975_v411_n44-46_p3894_Herrero2025-07-30T18:07:15Z Computing isolated roots of sparse polynomial systems in affine space Herrero, María Isabel Jeronimo, Gabriela Tali Sabia, Juan Vicente Rafael Algorithms Complexity Sparse polynomial systems Affine space Combinatorial structures Complexity Finite set Geometric resolution Homotopies Numerical algorithms Pre-processing Probabilistic algorithm Sparse polynomials System supports Systems of equations Algorithms Topology Polynomials We present a symbolic probabilistic algorithm to compute the isolated roots in Cn of sparse polynomial equation systems. As some already known numerical algorithms solving this task, our procedure is based on polyhedral deformations and homotopies, but it amounts to solving a smaller number of square systems of equations and in fewer variables. The output of the algorithm is a geometric resolution of a finite set of points including the isolated roots of the system. The complexity is polynomial in the size of the combinatorial structure of the system supports up to a pre-processing yielding the mixed cells in a subdivision of the family of these supports. © 2010 Elsevier B.V. All rights reserved. Fil:Herrero, M.I. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Jeronimo, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Sabia, J. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2010 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03043975_v411_n44-46_p3894_Herrero http://hdl.handle.net/20.500.12110/paper_03043975_v411_n44-46_p3894_Herrero
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Algorithms
Complexity
Sparse polynomial systems
Affine space
Combinatorial structures
Complexity
Finite set
Geometric resolution
Homotopies
Numerical algorithms
Pre-processing
Probabilistic algorithm
Sparse polynomials
System supports
Systems of equations
Algorithms
Topology
Polynomials
spellingShingle Algorithms
Complexity
Sparse polynomial systems
Affine space
Combinatorial structures
Complexity
Finite set
Geometric resolution
Homotopies
Numerical algorithms
Pre-processing
Probabilistic algorithm
Sparse polynomials
System supports
Systems of equations
Algorithms
Topology
Polynomials
Herrero, María Isabel
Jeronimo, Gabriela Tali
Sabia, Juan Vicente Rafael
Computing isolated roots of sparse polynomial systems in affine space
topic_facet Algorithms
Complexity
Sparse polynomial systems
Affine space
Combinatorial structures
Complexity
Finite set
Geometric resolution
Homotopies
Numerical algorithms
Pre-processing
Probabilistic algorithm
Sparse polynomials
System supports
Systems of equations
Algorithms
Topology
Polynomials
description We present a symbolic probabilistic algorithm to compute the isolated roots in Cn of sparse polynomial equation systems. As some already known numerical algorithms solving this task, our procedure is based on polyhedral deformations and homotopies, but it amounts to solving a smaller number of square systems of equations and in fewer variables. The output of the algorithm is a geometric resolution of a finite set of points including the isolated roots of the system. The complexity is polynomial in the size of the combinatorial structure of the system supports up to a pre-processing yielding the mixed cells in a subdivision of the family of these supports. © 2010 Elsevier B.V. All rights reserved.
author Herrero, María Isabel
Jeronimo, Gabriela Tali
Sabia, Juan Vicente Rafael
author_facet Herrero, María Isabel
Jeronimo, Gabriela Tali
Sabia, Juan Vicente Rafael
author_sort Herrero, María Isabel
title Computing isolated roots of sparse polynomial systems in affine space
title_short Computing isolated roots of sparse polynomial systems in affine space
title_full Computing isolated roots of sparse polynomial systems in affine space
title_fullStr Computing isolated roots of sparse polynomial systems in affine space
title_full_unstemmed Computing isolated roots of sparse polynomial systems in affine space
title_sort computing isolated roots of sparse polynomial systems in affine space
publishDate 2010
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03043975_v411_n44-46_p3894_Herrero
http://hdl.handle.net/20.500.12110/paper_03043975_v411_n44-46_p3894_Herrero
work_keys_str_mv AT herreromariaisabel computingisolatedrootsofsparsepolynomialsystemsinaffinespace
AT jeronimogabrielatali computingisolatedrootsofsparsepolynomialsystemsinaffinespace
AT sabiajuanvicenterafael computingisolatedrootsofsparsepolynomialsystemsinaffinespace
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