Ehrhart quasi-period collapse in rational polygons

TL;DR

This paper investigates the Ehrhart quasi-polynomials of rational polygons, focusing on the minimal periods of their coefficient functions, extending classical results from integral polygons to the rational case.

Contribution

It characterizes the possible minimal periods of Ehrhart quasi-polynomial coefficients for rational polygons, revealing new periodicity phenomena.

Findings

Identified possible minimal periods of Ehrhart quasi-polynomial coefficients.

Extended Ehrhart theory from integral to rational polygons.

Provided classifications of period behaviors in rational polygons.

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 *non*-integral convex polygons. Turning to the case in which the Ehrhart quasi-polynomial has nontrivial quasi-period, we determine the possible minimal periods of the coefficient functions of the Ehrhart quasi-polynomial of a rational polygon.