Computing generators of the ideal of a smooth affine algebraic variety

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Let K be an algebraically closed field, V ⊂ Kn be a smooth equidimensional algebraic variety and I (V) ⊂ K[x1,...,xn] be the ideal of all polynomials vanishing on V. We show that there exists a system of generators f1,...,fm of I (V) such that m ≤ (n - dim V) (1 + dim V) and deg(fi) ≤ deg V for i =...

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Autores principales: Blanco, C., Jeronimo, G., Solernó, P.
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: 2004
Materias:
Computation of the radical of a regular ideal
Efficient generation of polynomial ideals
Number and degree of generators of polynomial ideals
Regular signs
Straight-line programs
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_07477171_v38_n1_p843_Blanco
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paperaa:paper_07477171_v38_n1_p843_Blanco2023-06-12T16:48:11Z Computing generators of the ideal of a smooth affine algebraic variety J. Symb. Comput. 2004;38(1):843-872 Blanco, C. Jeronimo, G. Solernó, P. Computation of the radical of a regular ideal Efficient generation of polynomial ideals Number and degree of generators of polynomial ideals Regular signs Straight-line programs Let K be an algebraically closed field, V ⊂ Kn be a smooth equidimensional algebraic variety and I (V) ⊂ K[x1,...,xn] be the ideal of all polynomials vanishing on V. We show that there exists a system of generators f1,...,fm of I (V) such that m ≤ (n - dim V) (1 + dim V) and deg(fi) ≤ deg V for i = 1,...,m. If char(K) = 0 we present a probabilistic algorithm which computes the generators f1,..., fm from a set-theoretical description of V. If V is given as the common zero locus of s polynomials of degrees bounded by d encoded by straight-line programs of length L, the algorithm obtains the generators of I (V) with error probability bounded by E within complexity s(ndn)O(1)log2 (⌈1/E⌉)L. © 2004 Elsevier Ltd. All rights reserved. Fil:Blanco, C. 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:Solernó, P. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2004 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_07477171_v38_n1_p843_Blanco
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 Computation of the radical of a regular ideal
Efficient generation of polynomial ideals
Number and degree of generators of polynomial ideals
Regular signs
Straight-line programs
spellingShingle Computation of the radical of a regular ideal
Efficient generation of polynomial ideals
Number and degree of generators of polynomial ideals
Regular signs
Straight-line programs
Blanco, C.
Jeronimo, G.
Solernó, P.
Computing generators of the ideal of a smooth affine algebraic variety
topic_facet Computation of the radical of a regular ideal
Efficient generation of polynomial ideals
Number and degree of generators of polynomial ideals
Regular signs
Straight-line programs
description Let K be an algebraically closed field, V ⊂ Kn be a smooth equidimensional algebraic variety and I (V) ⊂ K[x1,...,xn] be the ideal of all polynomials vanishing on V. We show that there exists a system of generators f1,...,fm of I (V) such that m ≤ (n - dim V) (1 + dim V) and deg(fi) ≤ deg V for i = 1,...,m. If char(K) = 0 we present a probabilistic algorithm which computes the generators f1,..., fm from a set-theoretical description of V. If V is given as the common zero locus of s polynomials of degrees bounded by d encoded by straight-line programs of length L, the algorithm obtains the generators of I (V) with error probability bounded by E within complexity s(ndn)O(1)log2 (⌈1/E⌉)L. © 2004 Elsevier Ltd. All rights reserved.
format Artículo
Artículo
publishedVersion
author Blanco, C.
Jeronimo, G.
Solernó, P.
author_facet Blanco, C.
Jeronimo, G.
Solernó, P.
author_sort Blanco, C.
title Computing generators of the ideal of a smooth affine algebraic variety
title_short Computing generators of the ideal of a smooth affine algebraic variety
title_full Computing generators of the ideal of a smooth affine algebraic variety
title_fullStr Computing generators of the ideal of a smooth affine algebraic variety
title_full_unstemmed Computing generators of the ideal of a smooth affine algebraic variety
title_sort computing generators of the ideal of a smooth affine algebraic variety
publishDate 2004
url http://hdl.handle.net/20.500.12110/paper_07477171_v38_n1_p843_Blanco
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AT jeronimog computinggeneratorsoftheidealofasmoothaffinealgebraicvariety
AT solernop computinggeneratorsoftheidealofasmoothaffinealgebraicvariety
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