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arXiv · 2110.02893

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

Martin Henk·Stefan Kuhlmann·Robert Weismantel
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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.

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Martin Henk, Stefan Kuhlmann, Robert Weismantel. 2021-10-06. On lattice width of lattice-free polyhedra and height of Hilbert bases. https://arxiv.org/abs/2110.02893

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