Lattice polygons and the number 2i+7

TL;DR

This paper classifies convex lattice polygons based on their area and boundary/interior lattice points, establishing a key inequality and refining bounds using a new onion skin parameter.

Contribution

It provides a complete classification of lattice polygons with specific boundary and interior points, proving the inequality b ≤ 2i + 7 and introducing the onion skin parameter for sharper bounds.

Findings

Proved the inequality b ≤ 2i + 7 for lattice polygons.

Classified all triples (a, b, i) satisfying the polygon conditions.

Introduced the onion skin parameter to refine polygon bounds.

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.