On Intrinsic Bounds in the Nullstellensatz
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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 :=maxj deg( fj), then there exist polynomials p1, . . . , ps∈k[X1, . . . , Xn] such that 1=∑j pj fj and deg pj fj≦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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