Computing epsilon multiplicities in graded algebras

This article investigates the computational aspects of the $\varepsilon$-multiplicity. Primarily, we show that the $\varepsilon$-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 $\varepsilon$-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 $\varepsilon$-multiplicity in terms of mixed multiplicities by using the Veronese degree. This enabled us to make various explicit computations of $\varepsilon$-multiplicities. We further write a Macaulay2 algorithm to compute $\varepsilon$-multiplicity (under the Noetherian hypotheses) even when the base ring is not necessarily standard graded.