Computing generators of the ideal of a smooth affine algebraic variety

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, María Cristina, Jeronimo, Gabriela Tali, Solerno, Pablo Luis
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:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v38_n1_p843_Blanco 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

id	paper:paper_07477171_v38_n1_p843_Blanco
record_format	dspace
spelling	paper:paper_07477171_v38_n1_p843_Blanco2023-06-08T15:45:07Z Computing generators of the ideal of a smooth affine algebraic variety Blanco, María Cristina Jeronimo, Gabriela Tali Solerno, Pablo Luis 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 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v38_n1_p843_Blanco 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)
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, María Cristina Jeronimo, Gabriela Tali Solerno, Pablo Luis 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.
author	Blanco, María Cristina Jeronimo, Gabriela Tali Solerno, Pablo Luis
author_facet	Blanco, María Cristina Jeronimo, Gabriela Tali Solerno, Pablo Luis
author_sort	Blanco, María Cristina
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	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v38_n1_p843_Blanco http://hdl.handle.net/20.500.12110/paper_07477171_v38_n1_p843_Blanco
work_keys_str_mv	AT blancomariacristina computinggeneratorsoftheidealofasmoothaffinealgebraicvariety AT jeronimogabrielatali computinggeneratorsoftheidealofasmoothaffinealgebraicvariety AT solernopabloluis computinggeneratorsoftheidealofasmoothaffinealgebraicvariety
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Computing generators of the ideal of a smooth affine algebraic variety

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