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, ad, 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, ad−1, can be computed as follows: the lattice L induces a lattice LF on any face F of P; take the (d−1)-dimensional volume of F, divide by 2d(LF), 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.
Read more about this topic: Ehrhart Polynomial
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