Effective Nullstellensatz for arbitrary ideals

Let f_i be polynomials in n variables without a common zero. Hilbert's Nullstellensatz says that there are polynomials g_i such that \sum g_if_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.