“Ehrhart Polynomial”的概念、定义、习题解题方法及翻译-数学辞典

单词

Ehrhart Polynomial

释义

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 geometricinterpretations.

: 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 toPick'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 beinterpreted 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).

See also Dehn Invariant, Pick's Theorem

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.

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