Deformation Techniques for Efficient Polynomial Equation Solving

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Suppose we are given a parametric polynomial equation system encoded by an arithmetic circuit, which represents a generically flat and unramified family of zero-dimensional algebraic varieties. Let us also assume that there is given the complete description of the solution of a particular unramified...

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Autores principales: Heintz, J., Krick, T., Puddu, S., Sabia, J., Waissbein, A.
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: 2000
Materias:
Polynomial equation system; arithmetic circuit; shape (or primitive element) lemma; Newton-Hensel iteration
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_0885064X_v16_n1_p70_Heintz
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paperaa:paper_0885064X_v16_n1_p70_Heintz2023-06-12T16:48:17Z Deformation Techniques for Efficient Polynomial Equation Solving J. Complexity 2000;16(1):70-109 Heintz, J. Krick, T. Puddu, S. Sabia, J. Waissbein, A. Polynomial equation system; arithmetic circuit; shape (or primitive element) lemma; Newton-Hensel iteration Suppose we are given a parametric polynomial equation system encoded by an arithmetic circuit, which represents a generically flat and unramified family of zero-dimensional algebraic varieties. Let us also assume that there is given the complete description of the solution of a particular unramified parameter instance of our system. We show that it is possible to "move" the given particular solution along the parameter space in order to reconstruct - by means of an arithmetic circuit - the coordinates of the solutions of the system for an arbitrary parameter instance. The underlying algorithm is highly efficient, i.e., polynomial in the syntactic description of the input and the following geometric invariants: the number of solutions of a typical parameter instance and the degree of the polynomials occurring in the output. In fact, we prove a slightly more general result, which implies the previous statement by means of a well-known primitive element algorithm. We produce an efficient algorithmic description of the hypersurface obtained projecting polynomially the given generically flat family of varieties into a suitable affine space. © 2000 Academic Press. Fil:Krick, T. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Puddu, S. 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. 2000 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion application/pdf eng info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_0885064X_v16_n1_p70_Heintz
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
language Inglés
orig_language_str_mv eng
topic Polynomial equation system; arithmetic circuit; shape (or primitive element) lemma; Newton-Hensel iteration
spellingShingle Polynomial equation system; arithmetic circuit; shape (or primitive element) lemma; Newton-Hensel iteration
Heintz, J.
Krick, T.
Puddu, S.
Sabia, J.
Waissbein, A.
Deformation Techniques for Efficient Polynomial Equation Solving
topic_facet Polynomial equation system; arithmetic circuit; shape (or primitive element) lemma; Newton-Hensel iteration
description Suppose we are given a parametric polynomial equation system encoded by an arithmetic circuit, which represents a generically flat and unramified family of zero-dimensional algebraic varieties. Let us also assume that there is given the complete description of the solution of a particular unramified parameter instance of our system. We show that it is possible to "move" the given particular solution along the parameter space in order to reconstruct - by means of an arithmetic circuit - the coordinates of the solutions of the system for an arbitrary parameter instance. The underlying algorithm is highly efficient, i.e., polynomial in the syntactic description of the input and the following geometric invariants: the number of solutions of a typical parameter instance and the degree of the polynomials occurring in the output. In fact, we prove a slightly more general result, which implies the previous statement by means of a well-known primitive element algorithm. We produce an efficient algorithmic description of the hypersurface obtained projecting polynomially the given generically flat family of varieties into a suitable affine space. © 2000 Academic Press.
format Artículo
Artículo
publishedVersion
author Heintz, J.
Krick, T.
Puddu, S.
Sabia, J.
Waissbein, A.
author_facet Heintz, J.
Krick, T.
Puddu, S.
Sabia, J.
Waissbein, A.
author_sort Heintz, J.
title Deformation Techniques for Efficient Polynomial Equation Solving
title_short Deformation Techniques for Efficient Polynomial Equation Solving
title_full Deformation Techniques for Efficient Polynomial Equation Solving
title_fullStr Deformation Techniques for Efficient Polynomial Equation Solving
title_full_unstemmed Deformation Techniques for Efficient Polynomial Equation Solving
title_sort deformation techniques for efficient polynomial equation solving
publishDate 2000
url http://hdl.handle.net/20.500.12110/paper_0885064X_v16_n1_p70_Heintz
work_keys_str_mv AT heintzj deformationtechniquesforefficientpolynomialequationsolving
AT krickt deformationtechniquesforefficientpolynomialequationsolving
AT puddus deformationtechniquesforefficientpolynomialequationsolving
AT sabiaj deformationtechniquesforefficientpolynomialequationsolving
AT waissbeina deformationtechniquesforefficientpolynomialequationsolving
_version_ 1769810033805099008

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