2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76945Let $f_i$ 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 inequialities and to deformation theory are also discussed.LATEX2e, 25 pagesAlgebraic Geometrytext