Boundaries of pseudointegral polygons

We prove that a rational pseudointegral triangle with exactly one lattice point in its interior has at most $9$ lattice points on its boundary, where a polygon $P$ is called pseudointegral if the Ehrhart function of $P$ is a polynomial. We further show that such a triangle never has exactly $7$ lattice points on its boundary. Our results determine the set of all Ehrhart polynomials of rational triangles with one interior lattice point. In addition, we construct convex pseudointegral polygons with $i$ interior lattice points and $b$ boundary lattice points for all positive integral values of $(i,b)$ such that $b \le 5i + 4$. This is in contrast to integral polygons, which must satisfy $b \le 2i + 7$ by a result of Scott. Our constructions yield many new Ehrhart polynomials of rational polygons in the $i \ge 2$ case.