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Article: Inequalities and Ehrhart δvectors
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ABSTRACT: Fix a lattice L and let P be a ddimensional lattice polytope in L⊗ ℤ ℝ, i.e., the convex hull of finitely many points in L. The latticepoint counting function i(P,t):=#tP∩L is a polynomial in the positive integer variable r, by Ehrhart’s fundamental theorem [E. Ehrhart, C. R. Acad. Sci., Paris 254, 616–618 (1962; Zbl 0100.27601)]. Hence the generating function (the Ehrhart series) of i(P,r) is a rational function of the form ∑ t≥0 i(P,t)x t =δ d x d +δ d1 x d1 +⋯+δ 0 (1x) d+1 , and Stanley [R. P. Stanley, Ann. Discrete Math. 6, 333–342 (1980; Zbl 0812.52012)] proved that the δ k ’s are nonnegative integers. Let s be the degree of δ d x d +δ d1 x d1 +⋯+δ 0 and set l=d+1s and δ k =0 for k<0 or k>s. The paper under review gives several new inequalities for the δ k ’s, partly to improve on older results and partly to give purely combinatorial proofs for known results. For example, the new set of inequalities δ 2l +⋯+δ 0 +δ 1 ≤δ j +δ j1 +⋯+δ jl+1 (j=2,3,⋯,d1) generalizes T. Hibi’s [Adv. Math. 105, No. 2, 162–165 (1994; Zbl 0807.52011)] inequalities 1≤δ 1 ≤δ k (k=2,3,⋯,d1) which hold only when δ d ≠0 [M. Henk and M. Tagami, “Lower bounds on the coefficients of Ehrhart polynomials”, Eur. J. Comb. 30, No. 1, 70–83 (2009; Zbl 1158.52014)]. To mention one more sample theorem from the paper: suppose the boundary of P admits a regular unimodular lattice triangulation, then δ j+1 ≥δ dj and δ 0 +δ 1 +⋯+δ j+1 ≤δ d +δ d1 +⋯+δ dj +δ 1 δ d +j+1 j+1, for j=0,1,⋯,⌊d 2⌋1. These inequalities extend a theorem of C. Athanasiadis [Electron. J. Comb. 11, No. 2, Research paper R6, 13 p. (2004; Zbl 1068.52016)]. The proofs are based on a simple but powerful decomposition of the polynomial 1+t+⋯+t l1 δ d x d +δ d1 x d1 +⋯+δ 0 into two palindromic polynomials.Transactions of the American Mathematical Society 01/2009; 361(10). · 1.10 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: A condition is obtained on the Hilbert function of a graded CohenMacaulay domain over a field R0 = K when R is integral over the subalgebra generated by R1. A result of Eisenbud and Harris leads to a stronger condition when char K = 0 and R is generated as a Kalgebra by R1. An application is given to the Ehrhart polynomial of an integral convex polytope.Journal of Pure and Applied Algebra 08/1991; · 0.58 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We introduce a powerful connection between Ehrhart theory and additive number theory, and use it to produce infinitely many new classes of inequalities between the coefficients of the $h^*$polynomial of a lattice polytope. This greatly improves upon the three known classes of inequalities, which were proved using techniques from commutative algebra and combinatorics. As an application, we deduce all possible `balanced' inequalities between the coefficients of the $h^*$polynomial of a lattice polytope containing an interior lattice point, in dimension at most 6. Comment: 40 pages, 7 figures. Replaces `Kneser's theorem and inequalities in Ehrhart theory'. Improved dimension bounds04/2009;
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