The Ehrhart Polynomial of a Lattice Polytope

Diaz, Ricardo; Robins, Sinai

doi:10.2307/2951842

Cited by 87 publications

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“…In higher dimensions, the coefficient formulas similar to (17) have been given in [3,4]; and one can apply those coefficient formulas to an n-dimensional lattice simplex to obtain Zagier's reciprocity law of higher dimensional Dedekind sums; see [10,20]. However, I would like to mention another proof given by Beck [1] for the reciprocity law of Dedekind sums, using the generating functions of Ehrhart polynomials of [5]. Finally, it should be pointed out that the idea to realize the reciprocity law of Dedekind sums by a lattice simplex is from the work of Diaz and Robins [5], Kantor and Khovanskii [11], Pommersheim [15], though they employed more advanced tools.…”

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confidence: 99%

“…However, I would like to mention another proof given by Beck [1] for the reciprocity law of Dedekind sums, using the generating functions of Ehrhart polynomials of [5]. Finally, it should be pointed out that the idea to realize the reciprocity law of Dedekind sums by a lattice simplex is from the work of Diaz and Robins [5], Kantor and Khovanskii [11], Pommersheim [15], though they employed more advanced tools.…”

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The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

Chen

2003

Annals of Combinatorics

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This paper is to provide some new generalizations of the Pick Theorem. We first derive a point-set version of the Pick Theorem for an arbitrary bounded lattice polyhedron, then using the idea of weight function of [2] to obtain a weighted version; other Pick type theorems known to the author for integral lattice Z 2 are reduced to some special cases of the generalization. Finally, using the idea of Ehrhart [6] and the Pick Theorem, we give a direct proof of the reciprocity law for Dedekind sums. The ideas and methods presented here may be pushed to higher dimensions. Point-Set Version of the Pick TheoremLet P be a lattice polygon of R 2 , i.e., the vertices of P are points of the integral lattice Z 2 . Let i(P ) be the number of lattice points of P and i(∂P ) the number of lattice points of its boundary ∂σ. The Pick Theorem says thatGiven a bounded lattice polyhedron X of R 2 ; we denote byX the closure of X and by int X the interior of X; the frontier of X is the setX − intX. The link of X near a point x ∈X is the intersection of X and a circle S 1 (x, r) centered at x with small enough radius r; the Euler characteristic χ(lk (x, X)) is a local topological invariant, which plays an important role in our Pick type theorems. For an interior point x ∈ int X, the link lk (x, X) is a circle and has Euler characteristic zero. If X is closed, then for any x ∈ fr X, the link lk (x, X) is a collection of finite number of arcs and points, so χ(lk (x, X)) = the number of branches near x.

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The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

Chen

2003

Annals of Combinatorics

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“…Also the second leading coefficient admits a simple geometric interpretation as normalized surface area of P which we present in detail in (4.1). All other coefficients G i (P ), 1 ≤ i ≤ n − 2, have no such direct geometric meaning, except for special classes of polytopes (cf., e.g., [3,6,12,19,25,26,27,28,32]). …”

Section: Introductionmentioning

confidence: 99%

Notes on the Roots of Ehrhart Polynomials

Bey¹,

Henk²,

Wills

2007

Discrete Comput Geom

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Abstract. We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n 2 , where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun [8] where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n 2 .For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes.We further give a characterisation of the roots of the Ehrhart polyomials in the 3-dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part -1/2. These polytopes belong to the class of reflexive polytopes.

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“…However, the work of [16] used the Bott theorem of differential geometry to express these coefficients in terms of rational functions on Grassmanians; [3] and [11] used the RiemannRoch theorem of algebraic geometry to relate the Todd class and the Chern class of toric variety to express these coefficients in terms of Dedekind sums and cotangent functions; and the work of [2] used the technique of deformation for simple-polytopes and a combinatorial version of the Riemann-Roch theorem. The work of [8] is to express the generating functions of the sequences L(σ, m) and L(σ 0 , m) in terms of hyperbolic cotangent functions for a lattice simplex σ, whose vertices are the column vectors of a lower triangular matrix plus the origin. These generating functions are important because the coefficients of Ehrhart polynomials can be easily computed from their explicit expressions.…”

Section: Introductionmentioning

confidence: 99%

“…These generating functions are important because the coefficients of Ehrhart polynomials can be easily computed from their explicit expressions. The method of [8] is analytic, using the Poisson summation and the Fourier analysis, and involves some lengthy estimations. It seems that it is still an open problem to find a general formula with geometric interpretation for the coefficients of Ehrhart polynomials.…”

Section: Introductionmentioning

confidence: 99%

Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra

Chen¹

2002

Discrete Comput Geom

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Let σ be a simplex of R N with vertices in the integral lattice Z N . The number of lattice points of mσ (= {mα : α ∈ σ}) is a polynomial function L(σ, m) of m ≥ 0. In this paper we present: (i) a formula for the coefficients of the polynomial L(σ, t) in terms of the elementary symmetric functions; (ii) a hyperbolic cotangent expression for the generating functions of the sequence L(σ, m), m ≥ 0; (iii) an explicit formula for the coefficients of the polynomial L(σ, t) in terms of torsion. As an application of (i), the coefficient for the lattice n-simplex of R n with the vertices (0, . . . , 0, a j , 0, . . . , 0) (1 ≤ j ≤ n) plus the origin is explicitly expressed in terms of Dedekind sums; and when n = 2, it reduces to the reciprocity law about Dedekind sums. The whole exposition is elementary and self-contained.

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The Ehrhart Polynomial of a Lattice Polytope

Cited by 87 publications

References 14 publications

The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

Notes on the Roots of Ehrhart Polynomials

Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra

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