Ehrhart polynomials of integral simplices with prime volumes

Source: arXiv

ABSTRACT For an integral convex polytope $\Pc \subset \RR^N$ of dimension $d$, we call
$\delta(\Pc)=(\delta_0, \delta_1,..., \delta_d)$ the $\delta$-vector of $\Pc$
and $\vol(\Pc)=\sum_{i=0}^d\delta_i$ its normalized volume. In this paper, we
will establish the new equalities and inequalities on $\delta$-vectors for
integral simplices whose normalized volumes are prime. Moreover, by using
those, we will classify all the possible $\delta$-vectors of integral simplices
with normalized volume 5 and 7.

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    ABSTRACT: Fix a lattice L and let P be a d-dimensional lattice polytope in L⊗ ℤ ℝ, i.e., the convex hull of finitely many points in L. The lattice-point 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 d-1 x d-1 +⋯+δ 0 (1-x) 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 d-1 x d-1 +⋯+δ 0 and set l=d+1-s 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 2-l +⋯+δ 0 +δ 1 ≤δ j +δ j-1 +⋯+δ j-l+1 (j=2,3,⋯,d-1) generalizes T. Hibi’s [Adv. Math. 105, No. 2, 162–165 (1994; Zbl 0807.52011)] inequalities 1≤δ 1 ≤δ k (k=2,3,⋯,d-1) 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 d-j and δ 0 +δ 1 +⋯+δ j+1 ≤δ d +δ d-1 +⋯+δ d-j +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 l-1 δ d x d +δ d-1 x d-1 +⋯+δ 0 into two palindromic polynomials.
    Transactions of the American Mathematical Society 01/2009; 361(10). · 1.10 Impact Factor
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    ABSTRACT: A condition is obtained on the Hilbert function of a graded Cohen-Macaulay 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 K-algebra 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
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    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 bounds


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