Quasi-period collapse in half-integral polygons

TL;DR

This paper classifies Ehrhart polynomials of half-integral polygons exhibiting quasi-period collapse, revealing new polynomial forms and detailed classifications of polygons with specific lattice point properties.

Contribution

It provides a complete classification of Ehrhart polynomials for half-integral polygons with quasi-period collapse, including explicit polynomial examples and polygon classifications.

Findings

Classified Ehrhart polynomials for all such polygons.

Constructed new Ehrhart polynomials for rational polygons.

Identified exactly 30 polygons with specific interior lattice points.

Abstract

A half-integral polygon with quasi-period collapse behaves similarly to a lattice polygon in the sense that the number of lattice points in its integer dilates can be calculated as values of a polynomial, its Ehrhart polynomial. As a main result, we classify the Ehrhart polynomials of all half-integral non-lattice polygons with quasi-period collapse. In particular, we obtain that for any positive integer $i$, the polynomial $\frac{4i+5}{2}t^2+\frac{2i+7}{2}t+1\in \mathbb{Q}[t]$ is an Ehrhart polynomial of a rational polygon, which was an open question for $i>1$. We also study some extreme cases in detail. In particular, we show that up to affine unimodular equivalence there exist exactly $30$ half-integral non-lattice polygons with quasi-periodic collapse with exactly one interior lattice point, which are the dual polygons of the $30$ LDP polygons of Gorenstein index $2$. Furthermore, we classify all half-integral polygons with quasi-period collapse with at most $6$ interior lattice points or with $i\geq 1$ interior lattice points and the maximum possible number $2i+7$ of boundary lattice points.