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
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.
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 
,
.
Then 
is a polynomial function in 
of degree 
with rational coefficients
is the content of 
.
is half the sum of the contents of the 
-dimensional faces of 
.
.
denote the sum of the lattice lengths of the edges of 
, then the case 
corresponds to Pick's theorem,
denote the sum of the lattice volumes of the two-dimensional faces of 
, then the case 
gives
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,](/images/equations/EhrhartPolynomial/NumberedEquation5.svg)
is a Dedekind sum, 
, 
, 
(here, GCD is the greatest
common divisor), and 
(Pommersheim 1993).
