On the Upper bound of the Multiplicity Conjecture

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Let A = K[X₁,...,Xₙ] 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 Iᵏ and all k ≫ 0) if I belongs to any of the following large classes of ideals: enumerate[(1)] radical ideals. monomial ideals with generators in different degrees. zero-dimensional ideals with generators in different degrees. enumerate Surprisingly, our proof uses local techniques like analyticity, reductions, equimultiplicity and local results like Rees's theorem on multiplicities.