On the span of lattice points in a parallelepiped

TL;DR

This paper characterizes the linear functions vanishing on lattice points within a unit cube for a given lattice containing Z^n, providing explicit formulas for the span dimension, extending previous work on lattice point structures.

Contribution

It generalizes the Terminal Lemma to describe the space of linear functions vanishing on lattice points in a cube, with explicit dimension formulas.

Findings

Characterization of linear functions vanishing on lattice points in [0,1)^n.

Explicit formula for the dimension of the span of lattice points.

Generalization of the Terminal Lemma for broader lattice classes.

Abstract

Let $\Lambda\subset\mathbf{R}^{n}$ be a lattice which contains the integer lattice $\mathbf{Z}^{n}$. We characterize the space of linear functions $\mathbf{R}^{n}\rightarrow\mathbf{R}$ which vanish on the lattice points of $\Lambda$ lying in the half-open unit cube $[0,1)^{n}$. We also find an explicit formula for the dimension of the linear span of $\Lambda\cap[0,1)^{n}$. The results in this paper generalize and are based on the Terminal Lemma of Reid, which is in turn based upon earlier work of Morrison and Stevens on the classification of four dimensional isolated Gorenstein terminal cyclic quotient singularities.