Lattice Size of Width One Lattice Polytopes in ${\mathbb{R}}^3$

Abstract

The lattice size $\mathrm{l}{\mathrm{s}}_{\mathrm{\Delta }}(P)$ of a lattice polytope $P$ is a geometric invariant, which was formally introduced in relation to the problem of bounding the total degree and the bi-degree of the defining equation of an algebraic curve, but appeared implicitly earlier in geometric combinatorics. In this paper, we show that for an empty lattice polytope $P\subset {\mathbb{R}}^3$ there exists a reduced basis of ${\mathbb{Z}}^3$ which computes its lattice size $\mathrm{l}{\mathrm{s}}_{\mathrm{\Delta }}(P)$. This leads to a fast algorithm for computing $\mathrm{l}{\mathrm{s}}_{\mathrm{\Delta }}(P)$ for such $P$. We also extend this result to another class of lattice width one polytopes $P\subset {\mathbb{R}}^3$. We then provide a counterexample demonstrating that this result does not hold true for an arbitrary lattice polytope $P\subset {\mathbb{R}}^3$ of lattice width one.

15 pages, 4 figures

Cited by in corpus (1)

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