The number of integer points in will be counted by the quasi-polynomial Interpretation of coefficients If is closed (i.e. the boundary faces belong to ), some of the coefficients of have an easy interpretation: the leading coefficient, , is equal to the -dimensional volume of , divided by (see lattice for an explanation of the content or covolume of a lattice); the second coefficient, , can be computed as follows: the lattice induces a lattice on any face of ; take the -dimensional volume of , divide by , and add those numbers for all faces of ; the constant coefficient is the Euler characteristic of . When is a closed convex polytope, . The Betke–Kneser theorem Ulrich Betke and Martin Kneser established the following characterization of the Ehrhart coefficients. A functional defined on integral polytopes is an and translation invariant valuation if and only if there are real numbers such that Ehrhart series We can define a generating function for the Ehrhart polynomial of an integral -dimensional polytope as This series can be expressed as a rational function. Specifically, Ehrhart proved (1962) that there exist complex numbers , , such that the Ehrhart series of is Additionally, Richard P. Stanley's non-negativity theorem states that under the given hypotheses, will be non-negative integers, for . Another result by Stanley shows that if is a lattice polytope contained in , then for all . The -vector is in general not unimodal, but it is whenever it is symmetric, and the polytope has a regular unimodular triangulation. Ehrhart series for rational polytopes As in the case of polytopes with integer vertices, one defines the Ehrhart series for a rational polytope. For a d-dimensional rational polytope , where is the smallest integer such that is an integer polytope ( is called the denominator of ), then one has where the are still non-negative integers. Non-leading coefficient bounds The polynomial's non-leading coefficients in the representation can be upper bounded: where is a Stirling number of the first kind. Lower bounds also exist. Toric variety The case and of these statements yields Pick's theorem. Formulas for the other coefficients are much harder to get; Todd classes of toric varieties, the Riemann–Roch theorem as well as Fourier analysis have been used for this purpose. If is the toric variety corresponding to the normal fan of , then defines an ample line bundle on , and the