Abstract

If $P\subset \R^d$ is a rational polytope, then $i_P:=#$ is a quasi-polynomial in $n$, called the Ehrhart quasi-polynomial of $P$. The period of $i_P$ must divide $\LL= \min \{n \in \Z_{> 0} \colon nP \text{is an integral polytope}\}$. Few examples are known where the period is not exactly $\LL$. We show that for any $\LL$, there is a 2-dimensional triangle $P$ such that $\LL=\LL$ but such that the period of $i_P$ is 1, that is, $i_P$ is a polynomial in $n$. We also characterize all polygons $P$ such that $i_P$ is a polynomial. In addition, we provide a counterexample to a conjecture by T. Zaslavsky about the periods of the coefficients of the Ehrhart quasi-polynomial.