Article

Ehrhart polynomials of integral simplices with prime volumes

07/2011;
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.

0 Bookmarks
 · 
53 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: For any lattice polytope $P$, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known inequalities satisfied by the coefficients of the Ehrhart $\delta$-vector of a lattice polytope. We also provide combinatorial proofs of two results of Stanley that were previously established using techniques from commutative algebra. Finally, we give a necessary numerical criterion for the existence of a regular unimodular lattice triangulation of the boundary of a lattice polytope. Comment: 11 pages. v2: minor changes, more detailed proof of Lemma 2.12. To appear in Trans. Amer. Math. Soc
    01/2008;
  • Source
    [Show abstract] [Hide abstract]
    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; · 0.53 Impact Factor
  • Source
    [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 bounds
    04/2009;

Full-text (2 Sources)

Download
0 Downloads
Available from
Dec 8, 2014