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Computing epsilon multiplicities in graded algebras

Suprajo Das, Saipriya Dubey, Sudeshna Roy, Jugal K. Verma

TL;DR

It is shown that the $\varepsilon$-multiplicity of a homogeneous ideal in a two-dimensional standard graded domain of finite type over an algebraically closed field of arbitrary characteristic, is always a rational number.

Abstract

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.

Computing epsilon multiplicities in graded algebras

TL;DR

It is shown that the -multiplicity of a homogeneous ideal in a two-dimensional standard graded domain of finite type over an algebraically closed field of arbitrary characteristic, is always a rational number.

Abstract

This article investigates the computational aspects of the -multiplicity. Primarily, we show that the -multiplicity of a homogeneous ideal 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 in terms of certain mixed multiplicities associated to . In any dimension, under the assumptions that the saturated Rees algebra of 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.
Paper Structure (7 sections, 20 theorems, 153 equations)

This paper contains 7 sections, 20 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Epsilon multiplicity of monomial ideals
  4. Epsilon function in dimension one
  5. Hilbert polynomials of bigraded Rees algebras and mixed multiplicities
  6. Epsilon multiplicity in graded dimension two
  7. An algorithm for computing epsilon-multiplicities

Key Result

Theorem 2.3

hoang Assume that the ideal $I$ is generated in degrees $b_1\leq \cdots \leq b_s$. Then there exist integers $u_0\geq 0$ and $v_0\geq 0$ such that for all $u\geq b_sv + u_0$ and $v\geq v_0$, the bivariate Hilbert function $\dim_k \left(I^v\right)_u$ is equal to a bivariate numerical polynomial $P(u, then the coefficients $e_i(R[It])$ are integers for all $i=0,\ldots,d-1$ with $e_{d-1}(R[It]) = e(R

Theorems & Definitions (63)

  • Theorem 2.3
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • Example 4.4
  • ...and 53 more