On Intrinsic Bounds in the Nullstellensatz

Abstract. 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.