On the degrees of bases of free modules over a polynomial ring

Let k be an infinite field, A the polynomial ring k[x1 , . . . , xn] and F ∈ AN×M a matrix such that Im F ⊂ AN A-free (in particular, Quillen-Suslin Theorem implies that Ker F is also free). Let D be the maximum of the degrees of the entries of F and s the rank of F. We show that there exists a basi...

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Autores principales: Almeida, M., D'Alfonso, L., Solerno, P.
Formato: JOUR
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00255874_v231_n4_p679_Almeida
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id todo:paper_00255874_v231_n4_p679_Almeida
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spelling todo:paper_00255874_v231_n4_p679_Almeida2023-10-03T14:36:22Z On the degrees of bases of free modules over a polynomial ring Almeida, M. D'Alfonso, L. Solerno, P. Let k be an infinite field, A the polynomial ring k[x1 , . . . , xn] and F ∈ AN×M a matrix such that Im F ⊂ AN A-free (in particular, Quillen-Suslin Theorem implies that Ker F is also free). Let D be the maximum of the degrees of the entries of F and s the rank of F. We show that there exists a basis {v1 , . . . , vM} of AM such that {v1 , . . . , vM-s} is a basis of Ker F, {F(vM-s+1), . . . , F(vM)} is a basis of Im F and the degrees of their coordinates are of order ((M - s)sD)O(n4). This result allows to obtain a single exponential degree upper bound for a basis of the coordinate ring of a reduced complete intersection variety in Noether position. Fil:Almeida, M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Solerno, P. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00255874_v231_n4_p679_Almeida
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description Let k be an infinite field, A the polynomial ring k[x1 , . . . , xn] and F ∈ AN×M a matrix such that Im F ⊂ AN A-free (in particular, Quillen-Suslin Theorem implies that Ker F is also free). Let D be the maximum of the degrees of the entries of F and s the rank of F. We show that there exists a basis {v1 , . . . , vM} of AM such that {v1 , . . . , vM-s} is a basis of Ker F, {F(vM-s+1), . . . , F(vM)} is a basis of Im F and the degrees of their coordinates are of order ((M - s)sD)O(n4). This result allows to obtain a single exponential degree upper bound for a basis of the coordinate ring of a reduced complete intersection variety in Noether position.
format JOUR
author Almeida, M.
D'Alfonso, L.
Solerno, P.
spellingShingle Almeida, M.
D'Alfonso, L.
Solerno, P.
On the degrees of bases of free modules over a polynomial ring
author_facet Almeida, M.
D'Alfonso, L.
Solerno, P.
author_sort Almeida, M.
title On the degrees of bases of free modules over a polynomial ring
title_short On the degrees of bases of free modules over a polynomial ring
title_full On the degrees of bases of free modules over a polynomial ring
title_fullStr On the degrees of bases of free modules over a polynomial ring
title_full_unstemmed On the degrees of bases of free modules over a polynomial ring
title_sort on the degrees of bases of free modules over a polynomial ring
url http://hdl.handle.net/20.500.12110/paper_00255874_v231_n4_p679_Almeida
work_keys_str_mv AT almeidam onthedegreesofbasesoffreemodulesoverapolynomialring
AT dalfonsol onthedegreesofbasesoffreemodulesoverapolynomialring
AT solernop onthedegreesofbasesoffreemodulesoverapolynomialring
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