Relative volumes and minors in monomial subrings.
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 \(A(P)\).
scientific article; zbMATH DE number 2006004
scientific article