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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| Autores principales: | , , |
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_09381279_v8_n2_p125_Krick |
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| Sumario: | 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. |
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