Ehrhart polynomial

Remarks

The polynomial $E_P$ has degree $d$, and $E_P(0) = 1$. If $d = n$, the leading coefficient of $E_P$ is the volume of $P$. In general the leading coefficient is the volume of $P$ measured inside its affine hull, normalised by the lattice of integer points of that affine hull.

Example: for the unit square $P = [0,1]^2$, the dilate $tP = [0,t]^2$ contains $(t+1)^2$ integer points, so $E_P(t) = t^2 + 2t + 1$. For the triangle with vertices $(0,0)$, $(1,0)$, $(0,1)$, the dilate contains the points $(a,b) \in \Z^2$ with $a, b \ge 0$ and $a + b \le t$, so $E_P(t) = (t+1)(t+2)/2$. In both cases the leading coefficient is the area, $1$ and $1/2$.

Non-example: the segment $P = [0, 1/2]$ is a rational polytope but not a lattice polytope. Here $L_P(t) = \floor{t/2} + 1$, which is not a polynomial in $t$; it is only a quasi-polynomial of period $2$. So the hypothesis that the vertices are integer points is needed.

The generating function $\sum_{t \ge 0} E_P(t) z^t$ is a rational function of the form $h^*(z)/(1-z)^{d+1}$, where $h^*$ is a polynomial with nonnegative integer coefficients.