On intrinsic bounds in the Nullstellensatz

Let k be a field and f1 , . . . , fs be non constant polynomials in k[X1 , . . . , Xn] which generate the trivial ideal. In this paper we define an invariant associated to the sequence f1 , . . . , fs: the geometric degree of the system. With this notion we can show the following effective Nullstell...

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Detalles Bibliográficos
Autores principales:	Krick, T., Sabia, J., Solernó, P.
Formato:	JOUR
Materias:	Complete intersection polynomial ideals Effective Nullstellensatz Geometric degree Trace theory Functions Geometry Number theory Set theory Hilbert Nullstellensatz Polynomials
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_09381279_v8_n2_p125_Krick
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

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spelling	todo:paper_09381279_v8_n2_p125_Krick2023-10-03T15:48:46Z On intrinsic bounds in the Nullstellensatz Krick, T. Sabia, J. Solernó, P. Complete intersection polynomial ideals Effective Nullstellensatz Geometric degree Trace theory Functions Geometry Number theory Set theory Geometric degree Hilbert Nullstellensatz Trace theory Polynomials Let k be a field and f1 , . . . , fs be non constant polynomials in k[X1 , . . . , Xn] which generate the trivial ideal. In this paper we define an invariant associated to the sequence f1 , . . . , fs: the geometric degree of the system. With this notion we can show the following effective Nullstellensatz: if δ denotes the geometric degree of the trivial system f1 , . . .. , fs and d:= maxjdeg(fj), then there exist polynomials p1 , . . . , ps ∈ k[X1 , . . . , Xn] such that 1 = ∑jpjfjand deg pjfj ≦ 3n2δd. Since the number δ is always bounded by (d + 1)n-1, one deduces a classical single exponential upper bound in terms of d and n, but in some cases our new bound improves the known ones. Fil:Krick, T. 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. Fil:Solernó, 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_09381279_v8_n2_p125_Krick
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Complete intersection polynomial ideals Effective Nullstellensatz Geometric degree Trace theory Functions Geometry Number theory Set theory Geometric degree Hilbert Nullstellensatz Trace theory Polynomials
spellingShingle	Complete intersection polynomial ideals Effective Nullstellensatz Geometric degree Trace theory Functions Geometry Number theory Set theory Geometric degree Hilbert Nullstellensatz Trace theory Polynomials Krick, T. Sabia, J. Solernó, P. On intrinsic bounds in the Nullstellensatz
topic_facet	Complete intersection polynomial ideals Effective Nullstellensatz Geometric degree Trace theory Functions Geometry Number theory Set theory Geometric degree Hilbert Nullstellensatz Trace theory Polynomials
description	Let k be a field and f1 , . . . , fs be non constant polynomials in k[X1 , . . . , Xn] which generate the trivial ideal. In this paper we define an invariant associated to the sequence f1 , . . . , fs: the geometric degree of the system. With this notion we can show the following effective Nullstellensatz: if δ denotes the geometric degree of the trivial system f1 , . . .. , fs and d:= maxjdeg(fj), then there exist polynomials p1 , . . . , ps ∈ k[X1 , . . . , Xn] such that 1 = ∑jpjfjand deg pjfj ≦ 3n2δd. Since the number δ is always bounded by (d + 1)n-1, one deduces a classical single exponential upper bound in terms of d and n, but in some cases our new bound improves the known ones.
format	JOUR
author	Krick, T. Sabia, J. Solernó, P.
author_facet	Krick, T. Sabia, J. Solernó, P.
author_sort	Krick, T.
title	On intrinsic bounds in the Nullstellensatz
title_short	On intrinsic bounds in the Nullstellensatz
title_full	On intrinsic bounds in the Nullstellensatz
title_fullStr	On intrinsic bounds in the Nullstellensatz
title_full_unstemmed	On intrinsic bounds in the Nullstellensatz
title_sort	on intrinsic bounds in the nullstellensatz
url	http://hdl.handle.net/20.500.12110/paper_09381279_v8_n2_p125_Krick
work_keys_str_mv	AT krickt onintrinsicboundsinthenullstellensatz AT sabiaj onintrinsicboundsinthenullstellensatz AT solernop onintrinsicboundsinthenullstellensatz
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On intrinsic bounds in the Nullstellensatz

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