Lattice Size of Plane Convex Bodies

Abstract

The lattice size $\operatorname{ls}_{\Delta}(P)$ of a lattice polygon $P$ with respect to the standard simplex $\Delta$ was introduced and studied by Castryck and Cools in the context of simplification of the defining equation of an algebraic curve. Earlier, Schicho provided an "onion skins" algorithm for mapping a lattice polygon $P$ into a small integer multiple of the standard simplex, based on passing successively to the convex hull of the interior lattice points of $P$. Castryck and Cools showed that this algorithm computes the lattice size of $P$. In this paper we show that for a plane convex body $P$ a reduced basis of $\mathbb{Z}^2$ computes the lattice size. This provides a lattice reduction algorithm for computing the lattice size, which works for any convex body $P\subset\mathbb{R}^2$ and outperforms the "onion skins" algorithm in the case when $P$ is a lattice polygon.