Equidecomposable and weakly neighborly polytopes

A polytope is equidecomposable if all its triangulations have the same face numbers. For an equidecomposable polytope all minimal affine dependencies have an equal number of positive and negative coefficients. A subclass consists of the weakly neighborly polytopes, those for which every set of vertices is contained in a face of at most twice the dimension as the set. Theh-vector of every triangulation of a weakly neighborly polytope equals theh-vector of the polytope itself. Combinatorial properties of this class of polytopes are studied. Gale diagrams of weakly neighborly polytopes with few vertices are characterized in the spirit of the known Gale diagram characterization of Lawrence polytopes, a special class of weakly neighborly polytopes.     

Unimodality Questions for Integrally Closed Lattice Polytopes

It is a famous open question whether every integrally closed reflexive polytope has a unimodal Ehrhart δ -vector. We generalize this question to arbitrary integrally closed lattice polytopes and we prove unimodality for the δ -vector of lattice parallelepipeds. This is the first nontrivial class of integrally closed polytopes. Moreover, we suggest a new approach to the problem for reflexive polytopes via triangulations.     

The g-theorem matrices are totally nonnegative

The g-theorem proved by Billera, Lee, and Stanley states that a sequence is the g-vector of a simplicial polytope if and only if it is an M-sequence. For any d-dimensional simplicial polytope the face vector is gMd where Md is a certain matrix whose entries are sums of binomial coefficients. Björner found refined lower and upper bound theorems by showing that the (2×2)-minors of Md are nonnegative. He conjectured that all minors of Md are nonnegative and that is the result of this note.     

Ehrhart h ?-Vectors of Hypersimplices

We consider the Ehrhart h ^?-vector for the hypersimplex. It is well-known that the sum of the $h_{i}^{*}$ is the normalized volume which equals the Eulerian numbers. The main result is a proof of a conjecture by R.?Stanley which gives an interpretation of the $h^{*}_{i}$ coefficients in terms of descents and exceedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes.     

Gorenstein polytopes and their stringy E-functions

Inspired by ideas from algebraic geometry, Batyrev and the first named author have introduced the stringy E-function of a Gorenstein polytope. We prove that this a priori rational function is actually a polynomial, which is part of a conjecture of Batyrev and the first named author. The proof relies on a comparison result for the lattice point structure of a Gorenstein polytope P, a face F of P and the face of the dual Gorenstein polytope corresponding to F. In addition, we study joins of Gorenstein polytopes and introduce the notion of an irreducible Gorenstein polytope. We show how these concepts relate to the decomposition of nef-partitions.     

Facets and volume of Gorenstein Fano polytopes

It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension d is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension . Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.     

Ehrhart Polynomials of Matroid Polytopes and Polymatroids

We investigate properties of Ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that, for fixed rank, Ehrhart polynomials of matroid polytopes and polymatroids are computable in polynomial time. The proof relies on the geometry of these polytopes as well as a new refined analysis of the evaluation of Todd polynomials. In the second half we discuss two conjectures about the h ^*-vector and the coefficients of Ehrhart polynomials of matroid polytopes; we provide theoretical and computational evidence for their validity.     

Special simplices and Gorenstein toric rings

Athanasiadis [Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley, J. Reine Angew. Math., to appear.] studies an effective technique to show that Gorenstein sequences coming from compressed polytopes are unimodal. In the present paper we will use such the technique to find a rich class of Gorenstein toric rings with unimodal h-vectors arising from finite graphs.     

Higher integrality conditions, volumes and Ehrhart polynomials

A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of k-integral polytopes, where 0-integral is equivalent to integral. We show that the Ehrhart polynomial of a k-integral polytope P has the properties that the coefficients in degrees less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Minkowski Length of 3D Lattice Polytopes

We study the Minkowski length L(P) of a lattice polytope P, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing L(P) where P is a 3D lattice polytope. We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if Q, a subpolytope of P, is the Minkowski sum of L=L(P) lattice polytopes Q _i , each of Minkowski length 1, then the total number of interior lattice points of the polytopes Q ₁,??,Q _L is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case.     

Lattice polytopes of degree 2

A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly i>0 interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimensions. In particular, there is only a finite number of quadratic polynomials with fixed leading coefficient being the h^∗-polynomial of a lattice polytope.     

Intersecting all edges of convex polytopes by planes

A characterization theorem is given for 3-dimensional convex polytopes Q having the following property: There exists a polytope P, isomorphic to Q, all edges of which can be cut by a pair of planes that miss all its vertices. The result yields an affirmative solution of a conjecture of B. Grünbaum.     

Abelian group matrices over the p-adic and rational integers

Let G be a finite abelian group of order n. Let Z and Q denote the rational integers and rationals, respectively. A group matrix for G over Z (or Q) is an n-square matrix of the form Σ_{g ∈ G}a_gP(g), where a_g ∈ Z (or Q) and P is the regular representation of G so that P(g) is an n-square permutation matrix and P(gh) = P(g)P(h) for all g, h ∈ G. It is known that if M is an arbitrary positive definite unimodular matrix over Z then there exists a matrix A over Q such that M = A^τA, where τ denotes transposition. This paper proves that the exact analogue of this theorem holds if one demands that M and A be group matrices for G over Z and Q, respectively. Furthermore, if M is a group matrix for G over the p-adic integers then necessary and sufficient conditions are given for the existence of a group matrix A for G over the p-adic numbers such that M = A^τA.     

On the variance of random polytopes

A random polytope is the convex hull of uniformly distributed random points in a convex body K. A general lower bound on the variance of the volume and f-vector of random polytopes is proved. Also an upper bound in the case when K is a polytope is given. For polytopes, as for smooth convex bodies, the upper and lower bounds are of the same order of magnitude. The results imply a law of large numbers for the volume and f-vector of random polytopes when K is a polytope.     

Classifying smooth lattice polytopes via toric fibrations

We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n?2d+1. This gives a sharp answer, for this class of polytopes, to a question raised by V.V. Batyrev and B. Nill.     

Roots of Ehrhart polynomials arising from graphs

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots α of Ehrhart polynomials of polytopes of dimension D satisfy −D≤Re(α)≤D−1, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order d lie in the circle |z+\fracd4| ￡ \fracd4|z+\frac{d}{4}| \le \frac{d}{4} or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip -\fracD2 ￡ Re(a) ￡ \fracD2-1-\frac{D}{2} \leq \mathrm{Re}(\alpha) \leq \frac{D}{2}-1. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.     

Computation of the Highest Coefficients of Weighted Ehrhart Quasi-polynomials of Rational Polyhedra

This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A.?Barvinok in the unweighted case (i.e., h≡1). In contrast to Barvinok’s method, our method is local, obtains an approximation on the level of generating functions, handles the general weighted case, and provides the coefficients in closed form as step polynomials of the dilation. To demonstrate the practicality of our approach, we report on computational experiments which show that even our simple implementation can compete with state-of-the-art software.     

Lattice polytopes,Hecke operators,and the Ehrhart polynomial

Let P be a simple lattice polytope. We define an action of the Hecke operators on E(P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E(P). We also describe how the Brion–Vergne formula for E(P) transforms under the Hecke operators for nonsingular lattice polytopes P.      

Sums of sets of lattice points and unimodular coverings of polytopes

If P is a lattice polytope (that is, the convex hull of a finite set of lattice points in \({\mathbf{R}^n}\)), then every sum of h lattice points in P is a lattice point in the h-fold sumset hP. However, a lattice point in the h-fold sumset hP is not necessarily the sum of h lattice points in P. It is proved that if the polytope P is a union of unimodular simplices, then every lattice point in the h-fold sumset hP is the sum of h lattice points in P.