Lattice polygons and the number 2i+7

Details Lattice polygons and the number 2i+7 Lattice polygons and the number 2i+7

2004-06-10

arxiv.org/abs/math/0406224v3

Mathematics

Combinatorics

19 pages, 22 figures. To appear in the American Mathematical Monthly

Scientific paper

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.

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