Lattice Polygons and the Number
نویسندگان
چکیده
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.
منابع مشابه
A Fast Algorithm for Covering Rectangular Orthogonal Polygons with a Minimum Number of r-Stars
Introduction This paper presents an algorithm for covering orthogonal polygons with minimal number of guards. This idea examines the minimum number of guards for orthogonal simple polygons (without holes) for all scenarios and can also find a rectangular area for each guards. We consider the problem of covering orthogonal polygons with a minimum number of r-stars. In each orthogonal polygon P,...
متن کامل0 Statistics of lattice animals ( polyominoes ) and polygons
We have developed an improved algorithm that allows us to enumerate the number of site animals (polyominoes) on the square lattice up to size 46. Analysis of the resulting series yields an improved estimate, τ = 4.062570(8), for the growth constant of lattice animals and confirms to a very high degree of certainty that the generating function has a logarithmic divergence. We prove the bound τ >...
متن کاملOsculating and neighbour-avoiding polygons on the square lattice*
We study two simple modifications of self-avoiding polygons (SAPs). Osculating polygons (OP) are a super-set in which we allow the perimeter of the polygon to touch at a vertex. Neighbour-avoiding polygons (NAP) are only allowed to have nearest-neighbour vertices provided these are joined by the associated edge and thus form a sub-set of SAPs. We use the finite lattice method to count the numbe...
متن کاملOn convex lattice polygons
Let II be a convex lattice polygon with b boundary points and c (5 1) interior points. We show that for any given a , the number b satisfies b 5 2e + 7 , and identify the polygons for which equality holds. A lattice polygon II is a simple polygon whose vertices are points of the integral lattice. We let A = 4(11) denote the area of II , b{U) the number of lattice points on the boundary of II , ...
متن کاملMoving Out the Edges of a Lattice Polygon
We review previous work of (mainly) Koelman, Haase and Schicho, and Poonen and Rodriguez-Villegas on the dual operations of (i) taking the interior hull and (ii) moving out the edges of a two-dimensional lattice polygon. We show how the latter operation naturally gives rise to an algorithm for enumerating lattice polygons by their genus. We then report on an implementation of this algorithm, by...
متن کاملOn the Number of Convex Lattice Polygons
We prove that there are at most exp{cA 1 ^} different lattice polygons of area A. This improves a result of V. I. Arnol'd.
متن کامل