2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141113Let $A = K[X_1,...,X_n]$ and let $I$ be a graded ideal in $A$. We show that the upper bound of Multiplicity conjecture of Herzog, Huneke and Srinivasan holds asymptotically (i.e., for $I^k$ and all $k \gg 0$) if $I$ belongs to any of the following large classes of ideals: \begin{enumerate}[\rm (1)] \item radical ideals. \item monomial ideals with generators in different degrees. \item zero-dimensional ideals with generators in different degrees. \end{enumerate} Surprisingly, our proof uses local techniques like analyticity, reductions, equimultiplicity and local results like Rees's theorem on multiplicities.6 pages, Many typos corrected. An additional section on examples added. To appear in Proc. of AMSCommutative Algebra13H15, 13D02 (Primary) 13D40, 13A30 (Secondary)text