Computing epsilon multiplicities in graded algebras
arxiv(2024)
摘要
This article investigates the computational aspects of the
ε-multiplicity. Primarily, we show that the
ε-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 ε-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 ε-multiplicity in terms of mixed
multiplicities by using the Veronese degree. This enabled us to make various
explicit computations of ε-multiplicities. We further write a
Macaulay2 algorithm to compute ε-multiplicity (under the Noetherian
hypotheses) even when the base ring is not necessarily standard graded.
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