Effective Nullstellensatz for arbitrary ideals

Abstract.

Let f _i be polynomials in n variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials g _i such that ∑g _i f _i =1. The effective versions of this result bound the degrees of the g _i in terms of the degrees of the f _j . The aim of this paper is to generalize this to the case when the f _i are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.