Relative volumes and minors in monomial subrings.
type: Item
instance of: scholarly article
title: Relative volumes and minors in monomial subrings.
zbMATH Open document ID: 1051.52008
DOI: 10.1016/S0024-3795(03)00612-8
author: Cesar A. Escobar
author: José Martínez-Bernal
author: Rafael H. Villarreal
published in: Linear Algebra and its Applications
publication date: 2003-11-19T00:00:00Z
review text: If \(P\) is a \(d\)-dimensional lattice polytope in \(\mathbb{R}^n\), then the Ehrhart polynomial of \(P\) counts \(|\mathbb{Z}^n\cap iP|\) for integer \(i\geq 0\); the coefficient of its leading term is the relative volume \(\text{vol}(P)\) of \(P\) as defined (by the authors) to be \[ \text{vol}(P)= \lim_{i\to\infty}\, {|\mathbb{Z}^n\cap iP|\over i^d}, \] where \(i\) ranges over the integers. The authors prove that, if \({\mathcal A}= \{v_1,\dots, v_q\}\subseteq \mathbb{Z}^n\) lies in a hyperplane not containing the origin and \(P= \text{conv\,}{\mathcal A}\), then \[ \text{vol}(P)=| T(\mathbb{Z}^n/(v_2- v_1,\dots, v_q- v_1))|\lim_{i\to \infty} \,{|\mathbb{Z}{\mathcal A}\cap iP|\over i^d}, \] where the first term is the order of the torsion subgroup of the quotient of \(\mathbb{Z}^n\) by its subgroup generated by the \(v_i- v_1\). With \(K\) a field, \(x^a:= x^{a1}_1\cdots x^{a_n}_n\) for \(a_1,\dots, a_n\) nonnegative integers, and \(t\) a further indeterminate, let \(Ft:= \{x^at\mid a\in{\mathcal A}\}\), and \[ A(P):= K[x^a t^i\mid a\in\mathbb{Z}\cap iP,\,i\geq 0\text{ an integer}]. \] The authors then describe the conditions under which the integral closure of \(K[Ft]\) is \(A(P)\).
Mathematics Subject Classification ID: 52B20
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13B22
zbMATH DE Number: 2006004
zbMATH Keywords: lattice polytope
zbMATH Keywords: relative volume
zbMATH Keywords: Ehrhart ring
zbMATH Keywords: monomial ring
reviewed by: Peter McMullen
describes a project that uses: Normaliz
MaRDI profile type: Publication
cites work: Q5563439
cites work: Q4552815
cites work: Q4736149
cites work: Q4317713
cites work: Q4274014
cites work: Introduction to Toric Varieties. (AM-131)
cites work: Q5611879
cites work: Two theorems on matrices
cites work: Q3818127
cites work: On the ideal theory of graphs
cites work: Dimension and multiplicity for graded algebras
cites work: Hilbert functions of graded algebras
cites work: Q3748279
cites work: Q4200153
cites work: Q4861423
cites work: Monomial Algebras
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description: scientific article; zbMATH DE number 2006004