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)...
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_07477171_v38_n1_p843_Blanco https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_07477171_v38_n1_p843_Blanco_oai |
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I28-R145-paper_07477171_v38_n1_p843_Blanco_oai2024-08-16 Blanco, C. Jeronimo, G. Solernó, P. 2004 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. application/pdf http://hdl.handle.net/20.500.12110/paper_07477171_v38_n1_p843_Blanco info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar J. Symb. Comput. 2004;38(1):843-872 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 Computing generators of the ideal of a smooth affine algebraic variety info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_07477171_v38_n1_p843_Blanco_oai |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-145 |
| collection |
Repositorio Digital de la Universidad de Buenos Aires (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, 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 https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_07477171_v38_n1_p843_Blanco_oai |
| work_keys_str_mv |
AT blancoc computinggeneratorsoftheidealofasmoothaffinealgebraicvariety AT jeronimog computinggeneratorsoftheidealofasmoothaffinealgebraicvariety AT solernop computinggeneratorsoftheidealofasmoothaffinealgebraicvariety |
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1809357008611049472 |