Computing epsilon multiplicities in graded algebras

This article investigates the computational aspects of the -multiplicity. Primarily, we show that the -multiplicity of a homogeneous ideal I in a two-dimensional standard graded domain of finite type over an algebraically closed field of arbitrary characteristic, is always a rational number. In this situation, we produce a formula for the -multiplicity of I in terms of certain mixed multiplicities associated to I. In any dimension, under the assumptions that the saturated Rees algebra of I is finitely generated, we give a different expression of the -multiplicity in terms of mixed multiplicities by using the Veronese degree. This enabled us to make various explicit computations of -multiplicities. We further write a Macaulay2 algorithm to compute -multiplicity (under the Noetherian hypotheses) even when the base ring is not necessarily standard graded.