On Intrinsic Bounds in the Nullstellensatz |
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Authors: | T Krick J Sabia P Solernó |
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Affiliation: | (1) Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, -1428- Buenos Aires, Argentina (e-mail: krick@dm.uba.ar/jsabia@dm.uba.ar), AR;(2) Departamento de Economía y Matemática, Universidad de San Andrés, Vito Dumas 284, -1644- Victoria, Buenos Aires, Argentina (e-mail: psolerno@udesa.edu.ar), AR |
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Abstract: | Let k be a field and f
1, . . . , f
s
be non constant polynomials in kX
1, . . . , X
n
] which generate the trivial ideal. In this paper we define an invariant associated to the sequence f
1, . . . , f
s
: 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 f
1, . . . , f
s
and d :=max
j
deg( f
j
), then there exist polynomials p
1, . . . , p
s
∈kX
1, . . . , X
n
] such that 1=∑
j
p
j
f
j
and deg p
j
f
j
≦3n
2δd. 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.
Received November 24, 1995, revised version January 19, 1996 |
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Keywords: | : complete intersection polynomial ideals trace theory effective Nullstellensatz geometric degree |
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