Lattice Size of Plane Convex Bodies

TL;DR

This paper introduces a lattice reduction algorithm to compute the lattice size of any plane convex body, improving upon previous methods specifically for lattice polygons by leveraging reduced bases.

Contribution

It presents a new lattice reduction algorithm for calculating the lattice size of plane convex bodies, generalizing and outperforming the existing 'onion skins' algorithm for lattice polygons.

Findings

The new algorithm computes lattice size efficiently for any convex body.

It outperforms the 'onion skins' algorithm when applied to lattice polygons.

The method is based on reduced bases of .

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.