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In 1976, P. R. Scott characterized the Ehrhart polynomials of convex integralpolygons. We study the same question for Ehrhart polynomials andquasi-polynomials of \emph{non}-integral convex polygons. Define a\emph{pseudo-integral polygon}, or \emph{PIP}, to be a convex rational polygonwhose Ehrhart quasi-polynomial is a polynomial. The numbers of lattice pointson 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 Ehrhartquasi-polynomial has nontrivial quasi-period, we determine the possible minimalperiods that the coefficient functions of the Ehrhart quasi-polynomial of arational polygon may have.

Coefficient functions of the Ehrhart quasi-polynomials of rational polygons