Boundaries of pseudointegral polygons

TL;DR

This paper characterizes the boundary lattice points of rational pseudointegral triangles with one interior point, determines their Ehrhart polynomials, and constructs polygons with specific interior and boundary lattice point counts, expanding understanding of rational polygon Ehrhart theory.

Contribution

It proves bounds on boundary points of pseudointegral triangles with one interior point and constructs polygons with prescribed interior and boundary points, revealing new Ehrhart polynomial classes.

Findings

A rational pseudointegral triangle with one interior lattice point has at most 9 boundary points.

Such a triangle never has exactly 7 boundary lattice points.

Constructed polygons with all positive interior and boundary point counts satisfying b ≤ 5i + 4.

Abstract

We prove that a rational pseudointegral triangle with exactly one lattice point in its interior has at most $9$ lattice points on its boundary, where a polygon $P$ is called pseudointegral if the Ehrhart function of $P$ is a polynomial. We further show that such a triangle never has exactly $7$ lattice points on its boundary. Our results determine the set of all Ehrhart polynomials of rational triangles with one interior lattice point. In addition, we construct convex pseudointegral polygons with $i$ interior lattice points and $b$ boundary lattice points for all positive integral values of $(i,b)$ such that $b \le 5i + 4$. This is in contrast to integral polygons, which must satisfy $b \le 2i + 7$ by a result of Scott. Our constructions yield many new Ehrhart polynomials of rational polygons in the $i \ge 2$ case.