Lattice Polygons and the Number 2i + 7

In this note we classify all triples (a, b, i) such that there is a convex lattice polygon P with area a, and p respectively i lattice points on the boundary respectively in the interior. The crucial lemma for the classification is the necessity of b ≤ 2 i+7. We sketch three proofs of this fact: the original one by Scott [Sco76], an elementary one, and one using algebraic geometry. As a refinement, we introduce an onion skin parameter `: how many nested polygons does P contain? Then we use the “12” of Poonen and Villegas [PV00] to give sharper bounds.