Coefficient functions of the Ehrhart quasi-polynomials of rational polygons

TL;DR

This paper investigates the properties of Ehrhart quasi-polynomials of rational polygons, introducing pseudo-integral polygons (PIPs) with polynomial Ehrhart quasi-polynomials, and explores the possible periods of their coefficient functions.

Contribution

It characterizes PIPs with polynomial Ehrhart quasi-polynomials and determines possible minimal periods of coefficient functions for Ehrhart quasi-polynomials of rational polygons.

Findings

Existence of PIPs with boundary points 1 or 2 and arbitrary interior points

Open question on Scott's inequality for PIPs

Determination of minimal periods of coefficient functions

Abstract

In 1976, P. R. Scott characterized the Ehrhart polynomials of convex integral polygons. We study the same question for Ehrhart polynomials and quasi-polynomials of \emph{non}-integral convex polygons. Define a \emph{pseudo-integral polygon}, or \emph{PIP}, to be a convex rational polygon whose Ehrhart quasi-polynomial is a polynomial. The numbers of lattice points on the interior and on the boundary of a PIP determine its Ehrhart polynomial. We show that, unlike the integral case, there exist PIPs with $b=1$ or $b=2$ boundary points and an arbitrary number $I \ge 1$ of interior points. However, the question of whether a PIP must satisfy Scott's inequality $b \le 2I + 7$ when $I \ge 1$ remains open. Turning to the case in which the Ehrhart quasi-polynomial has nontrivial quasi-period, we determine the possible minimal periods that the coefficient functions of the Ehrhart quasi-polynomial of a rational polygon may have.