Lattice polygons and the number 2i+7
Combinatorics
2007-05-23 v3 Algebraic Geometry
Abstract
In this note we classify all triples (a,b,i) such that there is a convex lattice polygon P with area a, and b respectively i lattice points on the boundary respectively in the interior. The crucial lemma for the classification is the necessity of b \le 2 i + 7. We sketch three proofs of this fact: the original one by Scott, an elementary one, and one using algebraic geometry. As a refinement, we introduce an onion skin parameter l: how many nested polygons does P contain? and give sharper bounds.
Cite
@article{arxiv.math/0406224,
title = {Lattice polygons and the number 2i+7},
author = {Christian Haase and Josef Schicho},
journal= {arXiv preprint arXiv:math/0406224},
year = {2007}
}
Comments
19 pages, 22 figures. To appear in the American Mathematical Monthly