 TOPICS  # 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 (5)

where is a Dedekind sum, , , (here, GCD is the greatest common divisor), and (Pommersheim 1993).

Dehn Invariant, Pick's Theorem

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## References

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. 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.

## Referenced on Wolfram|Alpha

Ehrhart Polynomial

## Cite this as:

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