Coefficient functions of the Ehrhart quasi-polynomials of rational polygons

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 nonintegral convex polygons. Define a pseudo-integral polygon, or PIP, to be a convex rational polygon whose Ehrhart quasipolynomial 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 ≥ 1 of interior points. However, the question of whether a PIP must satisfy Scott’s inequality b ≤ 2I + 7 when I ≥ 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.