## Ehrhart Polynomial

Let denote an integral convex Polytope of Dimension in a lattice , and let denote the number of Lattice Points in dilated by a factor of the integer ,

 (1)

for . Then is a polynomial function in of degree with rational coefficients
 (2)

called the Ehrhart polynomial (Ehrhart 1967, Pommersheim 1993). Specific coefficients have important geometric interpretations.
1. is the Content of .

2. is half the sum of the Contents of the -D faces of .

3. .
Let denote the sum of the lattice lengths of the edges of , then the case corresponds to Pick's Theorem,
 (3)

Let denote the sum of the lattice volumes of the 2-D faces of , then the case gives
 (4)

where a rather complicated expression is given by Pommersheim (1993), since can unfortunately not be interpreted in terms of the edges of . The Ehrhart polynomial of the tetrahedron with vertices at (0, 0, 0), (, 0, 0), (0, , 0), (0, 0, ) is

 (5)
where is a Dedekind Sum, , , (here, gcd is the Greatest Common Divisor), and (Pommersheim 1993).

References

Ehrhart, E. Sur une problème de géométrie diophantine linéaire.'' J. Reine angew. Math. 227, 1-29, 1967.

MacDonald, I. G. The Volume of a Lattice Polyhedron.'' Proc. Camb. Phil. Soc. 59, 719-726, 1963.

McMullen, P. Valuations and Euler-Type Relations on Certain Classes of Convex Polytopes.'' Proc. London Math. Soc. 35, 113-135, 1977.

Pommersheim, J. Toric Varieties, Lattices Points, and Dedekind Sums.'' Math. Ann. 295, 1-24, 1993.

Reeve, J. E. On the Volume of Lattice Polyhedra.'' Proc. London Math. Soc. 7, 378-395, 1957.

Reeve, J. E. A Further Note on the Volume of Lattice Polyhedra.'' Proc. London Math. Soc. 34, 57-62, 1959.