On the upper bound of the multiplicity conjecture

Let A = K[X-1, ..., X-n] and let I be a graded ideal in A. We show that the upper bound of the multiplicity conjecture of Herzog, Huneke and Srinivasan holds asymptotically (i.e., for I-k and all k >> 0) if I belongs to any of the following large classes of ideals: ( 1) radical ideals, ( 2) monomial ideals with generators in different degrees, ( 3) zero-dimensional ideals with generators in different degrees. Surprisingly, our proof uses local techniques like analyticity, reductions, equimultiplicity and local results like Rees's theorem on multiplicities