[Paper Review] Lattice polygons and the number 2i+7

Christian Haase, Josef Schicho|ArXiv.org|Jun 10, 2004

Computational Geometry and Mesh Generation4 citations

TL;DR

This paper classifies all possible triples (a, b, i) of area, boundary lattice points, and interior lattice points for convex lattice polygons, proving that b ≤ 2i + 7 is necessary and sharp. It introduces the 'onion skin' parameter ℓ to refine this bound, showing (2ℓ−1)b ≤ 2i + 9ℓ² − 2, with equality only for multiples of the standard simplex Δ.

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.

Motivation & Objective

• To classify all triples (a, b, i) corresponding to convex lattice polygons, where a is area, b is boundary lattice points, and i is interior lattice points.
• To establish and refine the inequality b ≤ 2i + 7, originally due to Scott, as a necessary condition for the existence of such polygons.
• To introduce and analyze the 'onion skin' parameter ℓ, measuring the number of nested lattice polygons within a given polygon.
• To derive sharper bounds on b in terms of i and ℓ, improving upon Scott’s inequality by incorporating the hierarchical structure of lattice polygons.
• To explore connections between lattice polygon invariants and algebraic geometry, particularly via toric geometry and rational surfaces.

Proposed method

• Uses Pick’s Theorem (a = i + b/2 − 1) as a foundational identity to relate area, boundary, and interior points.
• Applies lattice equivalence transformations (SL₂(ℤ) ⋉ ℤ²) to normalize polygons into canonical forms for analysis.
• Employs a geometric 'onion skin' decomposition: recursively peeling off outer layers of lattice polygons to define ℓ, the number of nested convex lattice polygons.
• Derives bounds via case analysis on the height and base width of a polygon tightly fitted in a bounding box, using variables x = p/ℓ and y = (q + q′)/ℓ.
• Analyzes two quadratic forms p₁(x, y) = −xy + 4x + 2y − 9 and p₂(x, y) = −x² − xy + 8x + 4y − 18, showing that at least one is ≤ 0 for x ≥ 2, y ≥ 0, implying the main inequality.
• Provides three distinct proofs of b ≤ 2i + 7: Scott’s original, an elementary geometric proof, and an algebraic geometry proof via toric varieties.

Experimental results

Research questions

• RQ1 Which triples (a, b, i) can arise as the area, boundary lattice points, and interior lattice points of a convex lattice polygon?
• RQ2 What is the tightest possible upper bound on b in terms of i for convex lattice polygons, and can it be improved by incorporating additional invariants?
• RQ3 How does the 'onion skin' parameter ℓ, counting nested convex lattice polygons, refine the bound b ≤ 2i + 7?
• RQ4 Can the inequality b ≤ 2i + 7 be derived and generalized using algebraic geometry techniques, particularly toric geometry?
• RQ5 What is the algebraic geometry analogue of the onion-skin parameter ℓ, and does it yield new inequalities for rational surfaces?

Key findings

• The triple (a, b, i) arises from a convex lattice polygon if and only if b ≥ 3, i ≥ 0, a = i + b/2 − 1, and either i = 0, or i = 1 and b ≤ 9, or i ≥ 2 and b ≤ 2i + 6.
• The inequality b ≤ 2i + 7 is necessary for the existence of a convex lattice polygon with i interior points and b boundary points.
• The onion-skin parameter ℓ satisfies (2ℓ − 1)b ≤ 2i + 9ℓ² − 2, which is a strict improvement over b ≤ 2i + 7 when ℓ > 1.
• Equality in the refined bound (2ℓ − 1)b ≤ 2i + 9ℓ² − 2 holds only for multiples of the standard simplex Δ.
• The bound b ≤ 2i + 7 is sharp and achieved by specific polygons such as the triangle with i = 1, b = 9.
• The polynomial analysis shows that at least one of the two derived expressions p₁(x, y) or p₂(x, y) is non-positive for all x ≥ 2, y ≥ 0, confirming the inequality.