On Intrinsic Bounds in the Nullstellensatz

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Abstract.

 Let k be a field and  f 1, . . . ,  f s be non constant polynomials in k[X 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 k[X 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