Lattice size of 2D and 3D polytopes with respect to the cube
Anthony Harrison, Jenya Soprunova · arXiv (Cornell University) · 2017
We study the lattice size ${\rm ls}_\square(P)$ of a lattice polytope $P$ with respect to the unit cube $\square$. The lattice size ${\rm ls}_\square(P)$ is the smallest integer $l$ such that $P$ is contained in an $l$-dilate of the unit cube after some unimodular transformation $T$. A similar invariant, $\operatorname{ls_\Sigma}(P)$, where the unit cube is replaced with the standard simplex $\Sigma$, was studied by Schicho in the context of simplifying parametrizations of rational surfaces. Schicho gave an onion skins algorithm for mapping a lattice polygon $P$ into $l\Sigma$ for a small integer $l$. Castryck and Cools proved that this algorithm computes $\operatorname{ls_\Sigma}(P)$ and gave a similar algorithm for finding ${\rm ls}_\square(P)$ in the case when $P$ is a polygon. We provide a new algorithm for computing ${\rm ls}_\square(P)$ for a lattice polygon $P$, which does not require enumeration of lattice points in $P$. We also generalize our construction and explain a similar algorithm for computing the lattice size ${\rm ls}_\square(P)$ of 3D lattice polytopes.