1709.03454Sep 11, 2017The main result and the arguments have been reformulated in terms of lattice reduction. The results now apply to arbitrary convex bodies in $\mathbb{R}^2$, rather than lattice polygons as in the previous version
Lattice Size of Plane Convex Bodies
Anthony Harrison, Jenya Soprunova, Patrick Tierney
The lattice size lsΔ(P) of a lattice polygon P with respect to the standard simplex Δ 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
Nearby in the stack
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
Z2
computes the lattice size. This provides a lattice reduction algorithm for computing the lattice size, which works for any convex body
P⊂R2
and outperforms the "onion skins" algorithm in the case when