Deformation Techniques for Efficient Polynomial Equation Solving

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: Krick, Teresa Elena Genoveva, Puddu, Susana Isabel, Sabia, Juan Vicente Rafael
Publicado: 2000
Materias:
Polynomial equation system; arithmetic circuit; shape (or primitive element) lemma; Newton-Hensel iteration
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v16_n1_p70_Heintz
http://hdl.handle.net/20.500.12110/paper_0885064X_v16_n1_p70_Heintz
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paper:paper_0885064X_v16_n1_p70_Heintz2023-06-08T15:46:35Z Deformation Techniques for Efficient Polynomial Equation Solving Krick, Teresa Elena Genoveva Puddu, Susana Isabel Sabia, Juan Vicente Rafael 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 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v16_n1_p70_Heintz 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)
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
Krick, Teresa Elena Genoveva
Puddu, Susana Isabel
Sabia, Juan Vicente Rafael
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.
author Krick, Teresa Elena Genoveva
Puddu, Susana Isabel
Sabia, Juan Vicente Rafael
author_facet Krick, Teresa Elena Genoveva
Puddu, Susana Isabel
Sabia, Juan Vicente Rafael
author_sort Krick, Teresa Elena Genoveva
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 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v16_n1_p70_Heintz
http://hdl.handle.net/20.500.12110/paper_0885064X_v16_n1_p70_Heintz
work_keys_str_mv AT krickteresaelenagenoveva deformationtechniquesforefficientpolynomialequationsolving
AT puddususanaisabel deformationtechniquesforefficientpolynomialequationsolving
AT sabiajuanvicenterafael deformationtechniquesforefficientpolynomialequationsolving
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