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

Abstract

The lattice size $\operatorname{ls_\Delta}(P)$ 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 $\operatorname{ls_\Delta}(P)$. This leads to a fast algorithm for computing $\operatorname{ls_\Delta}(P)$ 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.