Lattice points in lattice polytopes

Betke, U.; McMullen, Peter

doi:10.1007/bf01312545

Cited by 87 publications

(69 citation statements)

References 8 publications

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“…Theorem 4 (see [5,23]). If P is a d-dimensional lattice point configuration and Γ is a unimodular triangulation of P , then h * conv(P ) (t) = t d h Γ ( 1 t ).…”

Section: Finding Growth Series From Unimodular Triangulationsmentioning

confidence: 95%

Root Polytopes and Growth Series of Root Lattices

Ardila¹,

Beck²,

Hoşten³

et al. 2011

SIAM J. Discrete Math.

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The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root lattices An, Cn, and Dn, and we compute their f -and h-vectors. This leads us to recover formulae for the growth series of these root lattices, which were first conjectured by Conway, Mallows, and Sloane and Baake and Grimm and were proved by Conway and Sloane and Bacher, de la Harpe, and Venkov. We also prove the formula for the growth series of the root lattice Bn, which requires a modification of our technique. Introduction.A lattice L is a discrete subgroup of R n for some n ∈ Z >0 . The rank of a lattice is the dimension of the subspace spanned by the lattice. We say that a lattice L is generated as a monoid by a finite collection of vectors M = {a 1 , . . . , a r } if each u ∈ L is a nonnegative integer combination of the vectors in M. For convenience, we often write the vectors from M as columns of a matrix M ∈ R n×r , and to make the connection between L and M more transparent, we refer to the lattice generated by M as L M . The word length of u with respect to M, denoted w(u), is min( c i ) taken over all expressions u = c i a i with c i ∈ Z ≥0 . The growth function S(k) counts the number of elements u ∈ L with word length w(u) = k with respect to M. We define the growth series to be the generating function G(x) := k≥0 S(k) x k . It is a

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“…Theorem 4 (see [5,23]). If P is a d-dimensional lattice point configuration and Γ is a unimodular triangulation of P , then h * conv(P ) (t) = t d h Γ ( 1 t ).…”

Section: Finding Growth Series From Unimodular Triangulationsmentioning

confidence: 95%

Root Polytopes and Growth Series of Root Lattices

Ardila¹,

Beck²,

Hoşten³

et al. 2011

SIAM J. Discrete Math.

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“…The following inequalities and their proof are a slight generalisation of [5,Theorem 6]. Recall that the Stirling number S i (d) of the first kind is the coefficient of…”

Section: Inequalities Between Coefficients Of Polynomialsmentioning

confidence: 99%

“…In the case when P is a d-dimensional lattice polytope and h(t) = δ P (t), the existence of the following inequalities was suggested by Betke and McMullen [5].…”

Section: Lemma 29mentioning

confidence: 99%

“…If we use the notation p(t) = a(t)(1 + t + · · · + t n−1 ) d+1 = n(d+1)−1 k=0 p k t k , thenã(t) = E n p(t) . (5) Observe that the symmetry of a(t) implies that p(t) = t n(d+1)−1 p( 1 t ). Hence, we may write p(t) = p 0 + p 1 t + · · · + p n t n + · · · + p n−1 t nd + · · · + p 1 t n(d+1)−2 + p 0 t n(d+1)−1 , which implies a(t) = p 0 + p n t + p 2n t 2 + · · · + p d 2 n t d 2 + p d+1 2 n−1 t d 2 +1 + · · · + p 2n−1 t d−1 + p n−1 t d .…”

Section: Lemma 41mentioning

confidence: 99%

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On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series

Beck

Stapledon

2008

Math. Z.

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For every positive integer n, consider the linear operator U n on polynomials of degree at most d with integer coefficients defined as follows: if we write h(t)(We show that there exists a positive integer n d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n d , U n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen-Macauley graded rings.

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“…It follows that δ Q (t) is a polynomial of degree less than or equal to d with non-negative coefficients and constant term 1 [4]. If Σ is complete, we conclude from Lemma 2.4 that δ Q (t) is a polynomial of degree d with positive integer coefficients [4]. Corollary 2.9.…”

Section: Substituting Inmentioning

confidence: 85%

Weighted Ehrhart theory and orbifold cohomology

Stapledon

2008

Advances in Mathematics

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We introduce the notion of a weighted δ-vector of a lattice polytope. Although the definition is motivated by motivic integration, we study weighted δ-vectors from a combinatorial perspective. We present a version of Ehrhart Reciprocity and prove a change of variables formula. We deduce a new geometric interpretation of the coefficients of the Ehrhart δ-vector. More specifically, they are sums of dimensions of orbifold cohomology groups of a toric stack.

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Lattice points in lattice polytopes

Cited by 87 publications

References 8 publications

Root Polytopes and Growth Series of Root Lattices

Root Polytopes and Growth Series of Root Lattices

On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series

Weighted Ehrhart theory and orbifold cohomology

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