Effectiveness of an affine invariant for indecomposable integral polytopes
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 that tK is projectively faithful, and if \(R_{tK}\neq R_{t'K}\), then \((R^+_{t'K}\cap R_{tK})\setminus R^+_{tK}\neq \emptyset.\) This gives insight into certain order- theoretic random walk problems. Moreover, if \(\phi (R_{t'K})\subseteq R_{tK}\) for some automorphism \(\phi\) of \(R_{dK}\), the \(R_{tK}=R_{t'K}\). Examples of integral polytopes of dimension \(\geq 5\) with very strange properties are presented.
scientific article; zbMATH DE number 4203168