Ehrhart Polynomial -- from Wolfram MathWorld

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 -dimensional 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 two-dimensional 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

l_Delta(k)=1/6abck^3+1/4(ab+ac+bc+d)k^2+[1/(12)((ac)/b+(bc)/a+(ab)/c+(d^2)/(abc))+1/4(a+b+c+A+B+C)-As((bc)/d,(aA)/d)-Bs((ac)/d,(bB)/d)-Cs((ab)/d,(cC)/d)]k+1,

(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.

Ellis-Monaghan, J. A. and Merino, C. "Graph Polynomials and Their Applications II: Interrelations and Interpretations." 28 Jun 2008. http://arxiv.org/abs/0806.4699.

Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 215, 1984.

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.

CITE THIS AS:

Weisstein, Eric W. "Ehrhart Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/EhrhartPolynomial.html