Ehrhart Polynomial - Interpretation of Coefficients

Interpretation of Coefficients

If P is closed (i.e. the boundary faces belong to P), some of the coefficients of L(P, t) have an easy interpretation:

• the leading coefficient, a_d, is equal to the d-dimensional volume of P, divided by d(L) (see lattice for an explanation of the content or covolume d(L) of a lattice);
• the second coefficient, a_d−1, can be computed as follows: the lattice L induces a lattice L_F on any face F of P; take the (d−1)-dimensional volume of F, divide by 2d(L_F), and add those numbers for all faces of P;
• the constant coefficient a₀ is the Euler characteristic of P. When P is a closed convex polytope, a₀ = 1.