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Boundaries of pseudointegral polygons

Tyrrell B. McAllister, Jason S. Williford

TL;DR

The paper establishes sharp boundary-point bounds for rational triangular PIPs with a single interior lattice point, showing the boundary count must satisfy $1\le b\le 9$ and $b\neq 7$, by reducing to a number-theoretic condition $b=(x+y+z)^2/(xyz)$ and proving the bound via Vieta jumping. It further constructs infinitely many pseudoreflexive triangles realizing every allowed $b$, including Fibonacci-based families with $b=9$, and proves the existence of infinite families of PRPs with $i=1$ and prescribed $b$. Beyond the triangular case, the authors build rational polygonal PIPs with $i\ge1$ interior points and $b\le 5i+4$ boundary points, yielding many new Ehrhart polynomials and demonstrating a broader range of realizable Ehrhart polynomials for rational polygons. These results extend the understanding of how boundary structure constrains Ehrhart polynomials in the non-integral setting and provide constructive families that push beyond classical integral-polygon bounds (Scott) and half-integral cases (Bohnert).

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.

Boundaries of pseudointegral polygons

TL;DR

The paper establishes sharp boundary-point bounds for rational triangular PIPs with a single interior lattice point, showing the boundary count must satisfy and , by reducing to a number-theoretic condition and proving the bound via Vieta jumping. It further constructs infinitely many pseudoreflexive triangles realizing every allowed , including Fibonacci-based families with , and proves the existence of infinite families of PRPs with and prescribed . Beyond the triangular case, the authors build rational polygonal PIPs with interior points and boundary points, yielding many new Ehrhart polynomials and demonstrating a broader range of realizable Ehrhart polynomials for rational polygons. These results extend the understanding of how boundary structure constrains Ehrhart polynomials in the non-integral setting and provide constructive families that push beyond classical integral-polygon bounds (Scott) and half-integral cases (Bohnert).

Abstract

We prove that a rational pseudointegral triangle with exactly one lattice point in its interior has at most lattice points on its boundary, where a polygon is called pseudointegral if the Ehrhart function of is a polynomial. We further show that such a triangle never has exactly 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 interior lattice points and boundary lattice points for all positive integral values of such that . This is in contrast to integral polygons, which must satisfy by a result of Scott. Our constructions yield many new Ehrhart polynomials of rational polygons in the case.
Paper Structure (8 sections, 29 theorems, 42 equations, 2 figures)

This paper contains 8 sections, 29 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pseudoreflexive polygons
  4. Counting boundary points of triangular PIPs
  5. Number-theoretic lemmas
  6. Infinite families of pseudoreflexive triangles
  7. New Ehrhart polynomials
  8. A generalization of the number-theory lemma

Key Result

Theorem 1.1

Let $T \subseteq \mathbb{R}^{2}$ be a rational pseudointegral triangle that contains exactly one lattice point in its interior, and let $b$ be the number of lattice points on the boundary of $T$. Then ${1 \le b \le 9}$, but $b \ne 7$.

Figures (2)

  • Figure 1: The $16$ reflexive polygons (adapted from HaaSch2009).
  • Figure 2: PIPs with $1$ or $2$ boundary lattice points.

Theorems & Definitions (59)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Example 1.4
  • Theorem 1.5: Sco1976
  • Theorem 1.6: Boh2024preprint
  • Theorem 1.7
  • Theorem 2.1: Ehrhart Ehr1962a
  • Theorem 2.2: Ehrhart--Macdonald reciprocity
  • Theorem 2.3
  • ...and 49 more