Toric varieties, lattice points and Dedekind sums

Pommersheim, James

doi:10.1007/bf01444874

Cited by 122 publications

(95 citation statements)

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“…We remark that for three-dimensional polytopes P, the only coefficient of the Ehrhart polynomial L(P, t) = c 3 t 3 +c 2 t 2 +c 1 t+1 that is difficult to determine is the codimension two coefficient c 1 (see [12]). This coefficient can be found by solving for the coefficient of s −2 in the Laurent expansion of the product of cotangents given by the corollary.…”

Section: Corollary L(p Int T) Is a Polynomial In T ∈ N And Ifmentioning

confidence: 99%

“…The other coefficients of L(P, t) remained a mystery, even for a general lattice 3-simplex, until the recent work of Pommersheim [12] in R 3 , Kantor and Khovanskii [6] in R 4 , and most recently Cappell and Shaneson [2] in R n . [12] and [6] used techniques from algebraic geometry related to the Todd classes of toric varieties to express these coefficients in terms of Dedekind sums and other cotangent expansions, and [2] used Grothendieck-Riemann-Roch for their work on convex lattice polytopes. The present paper introduces Fourier methods into the study of lattice polytopes, and some known recent results are shown to be easy corollaries of the main theorem.…”

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Diaz

Robins

1996

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. The problem of counting the number of lattice points inside a lattice polytope in R n has been studied from a variety of perspectives, including the recent work of Pommersheim and Kantor-Khovanskii using toric varieties and Cappell-Shaneson using Grothendieck-Riemann-Roch. Here we show that the Ehrhart polynomial of a lattice n-simplex has a simple analytical interpretation from the perspective of Fourier Analysis on the n-torus. We obtain closed forms in terms of cotangent expansions for the coefficients of the Ehrhart polynomial, that shed additional light on previous descriptions of the Ehrhart polynomial.The number of lattice points inside a convex lattice polytope in R n (a polytope whose vertices have integer coordinates) has been studied intensively by combinatorialists, algebraic geometers, number theorists, Fourier analysts, and differential geometers. Algebraic geometers have shown that the Hilbert polynomials of toric varieties associated to convex lattice polytopes precisely describe the number of lattice points inside their dilates [3]. Number theorists have estimated lattice point counts inside symmetric bodies in R n to get bounds on ideal norms and class numbers of number fields. Fourier analysts have estimated the number of lattice points in simplices using Poisson summation (see Siegel's classic solution of the Minkowski problem [14] and Randol's estimates for lattice points inside dilates of general planar regions [13]). Differential geometers have also become interested in lattice point counts in polytopes in connection with the Durfree conjecture [18].Let Z n denote the n-dimensional integer lattice in R n , and let P be an ndimensional lattice polytope in R n . Consider the function of an integer-valued variable t that describes the number of lattice points that lie inside the dilated polytope tP: L(P, t) = the cardinality of {tP} ∩ Z n .Ehrhart [4] inaugurated the systematic study of general properties of this function by proving that it is always a polynomial in t ∈ N, and that in fact L(P, t) = Vol(P)t n + 1 2 Vol(∂P)t n−1 + · · · + χ(P)

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Section: Corollary L(p Int T) Is a Polynomial In T ∈ N And Ifmentioning

confidence: 99%

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confidence: 99%

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Diaz

Robins

1996

Electron. Res. Announc. Amer. Math. Soc.

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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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“…If P is not compact, the combinatorial Euler characteristic is usually different from that defined by the alternating sum of Betti numbers; while if P is compact, they are the same; see [4] and [12] for example. The other coefficients of Ehrhart polynomials are still mysterious, even for a general lattice 3-simplex, until the recent work of Morelli [16] in R n , Pommersheim [18] in R 3 , Kantor and Khovanskii [11] in R 4 , Cappell and Shaneson [3] in R n , Brion and Vergne [2] in R n , and Diaz and Robins [8] in R n . For instance, the coefficients for a lattice tetrahedron of R 3 with vertices (0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c) are completely determined; Kanor and Khovanskii [11] gave a complete description of the codimension 2 coefficients of the 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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Toric varieties, lattice points and Dedekind sums

Cited by 122 publications

References 7 publications

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Notes on the Roots of Ehrhart Polynomials

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

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