Effectiveness of an affine invariant for indecomposable integral polytopes
type: Item
instance of: scholarly article
title: Effectiveness of an affine invariant for indecomposable integral polytopes
zbMATH Open document ID: 0728.52009
DOI: 10.1016/0022-4049(90)90083-T
published in: Journal of Pure and Applied Algebra
publication date: 1990-01-01T00:00:00Z
review text: An integral polytope K is a compact, convex polytope all of whose vertices lie in \({\mathbb{Z}}^ d\). Translation by some \(z\in {\mathbb{Z}}^ d\), or application of \(M\in GL(d,{\mathbb{Z}})\) (the group of \(d\times d\) matrices with integer entries having \(\det =\pm 1)\) yields a map of \({\mathbb{Z}}^ d\) to itself. Such maps generate the group AGL(d,\({\mathbb{Z}})\) of translations acting on integral polytopes K. K is projectively faithful if the group generated by \(K\cap {\mathbb{Z}}^ d-K\cap {\mathbb{Z}}^ d\) is the standard copy of \({\mathbb{Z}}^ d\) inside \({\mathbb{R}}^ d\). K is indecomposable if \(K=K_ 1+K_ 2\), \(K_ i\) real, implies that one summand must equal \(\lambda K+z\), where \(\lambda\in {\mathbb{R}}\), \(z\in {\mathbb{R}}^ d\). Denote the Laurent polynomial ring in the variables \(\{x_ i: i=1,2,...,d\}\), with \(x^ w=x_ 1^{w(1)}\cdot x_ 2^{w(2)}\cdot...\cdot x_ d^{w(d)}\) for monomials, by \({\mathbb{R}}[x_ i^{\pm 1}]\). Denote the set of Laurent polynomials with no negative coefficients by \({\mathbb{R}}[x_ i^{\pm 1}]^+\). If K is projectively faithful, set \(P=\Sigma \lambda_ wx^ w\) in \({\mathbb{R}}[x_ i^{\pm 1}]\), with \(w\in K\cap {\mathbb{Z}}^ d\), \(\lambda_ w\in {\mathbb{R}}\). Define \(R_ p\) as the real algebra generated by \(\{x^ w/P:\) \(w\in K\cap {\mathbb{Z}}^ d\}\), with the positive cone \(R^+_ p\) generated additively and multiplicatively by the same set of rational functions. Let U denote the set of elements in \(R_ p\) that are bounded below (away from zero) as functions on \((R^ d)^{++}\). Define \(R_ k\) to be \(R_ p[U^{-1}].\) The author's main new result is that K, \(K'\) projectively faithful indecomposable integral polytopes and \(R_ K\cong R_{K'}\) together imply that there exist positive integers m and n such that mK and \(nK'\) are AGL(d,\({\mathbb{Z}})\)-equivalent. Regarding the fine structure of \(R_ K\), for indecomposable K, he proves: if \(t<t'\) are positive integers and K is an integral polytope such…
Mathematics Subject Classification ID: 52B20
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 4203168
zbMATH Keywords: lattice polytope
zbMATH Keywords: affine equivalence
zbMATH Keywords: indecomposable integral polytopes
author: David Handelman
MaRDI profile type: Publication
cites work: Positive polynomials and product type actions of compact groups
cites work: Integral body-building in \({\mathbb{R}}^ 3\)
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cites work: Positive polynomials, convex integral polytopes, and a random walk problem
Wikidata QID: Q126340170
description: scientific article; zbMATH DE number 4203168