On lattice width of lattice-free polyhedra and height of Hilbert bases

TL;DR

This paper investigates the lattice width of lattice-free polyhedra, linking it to Hilbert bases and finite abelian groups, and provides bounds and characterizations that advance understanding in discrete geometry.

Contribution

It establishes a connection between lattice width, Hilbert bases, and group diameters, and proves the Hilbert basis conjecture for specific cases, offering new bounds for simplices.

Findings

Bound on lattice width of pyramids depending on Δ(A)

Complete characterization of Hilbert bases when Δ(A)=2

Validity of the Hilbert basis conjecture for simplicial cones

Abstract

We study the lattice width of lattice-free polyhedra given by $\mathbf{A}\mathbf{x}\leq\mathbf{b}$ in terms of $\Delta(\mathbf{A})$, the maximal $n\times n$ minor in absolute value of $\mathbf{A}\in\mathbb{Z}^{m\times n}$. Our main contribution is to link the lattice width of lattice-free polyhedra to the height of Hilbert bases and to the diameter of finite abelian groups. This leads to a bound on the lattice width of lattice-free pyramids which solely depends on $\Delta(\mathbf{A})$ provided a conjecture regarding the height of Hilbert bases holds. Further, we exploit a combination of techniques to obtain novel bounds on the lattice width of simplices. A second part of the paper is devoted to a study of the above mentioned Hilbert basis conjecture. We give a complete characterization of the Hilbert basis if $\Delta(\mathbf{A}) = 2$ which implies the conjecture in that case and prove its validity for simplicial cones.