Ehrhart Polynomial

Let Delta denote an integral convex polytope of dimension n in a lattice M, and let l_Delta(k) denote the number of lattice points in Delta dilated by a factor of the integer k,

 l_Delta(k)=#(kDelta intersection M)

for k in Z^+. Then l_Delta is a polynomial function in k of degree n with rational coefficients


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

1. a_n is the content of Delta.

2. a_(n-1) is half the sum of the contents of the (n-1)-dimensional faces of Delta.

3. a_0=1.

Let S_2(Delta) denote the sum of the lattice lengths of the edges of Delta, then the case n=2 corresponds to Pick's theorem,


Let S_3(Delta) denote the sum of the lattice volumes of the two-dimensional faces of Delta, then the case n=3 gives


where a rather complicated expression is given by Pommersheim (1993), since a_1 can unfortunately not be interpreted in terms of the edges of Delta. The Ehrhart polynomial of the tetrahedron with vertices at (0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c) is


where s(x,y) is a Dedekind sum, A=GCD(b,c), B=GCD(a,c), C=GCD(a,b) (here, GCD is the greatest common divisor), and d=ABC (Pommersheim 1993).

See also

Dehn Invariant, Pick's Theorem

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Beck, M. and Robins, S. Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. New York: Springer, 2007.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., 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.

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Ehrhart Polynomial

Cite this as:

Weisstein, Eric W. "Ehrhart Polynomial." From MathWorld--A Wolfram Web Resource.

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