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.     

h-Vectors of Gorenstein polytopes

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP, c∈N, can be written as the sum of c lattice points in P. (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies.     

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.     

A theorem about antiprisms

Let P be a polytope in Rⁿ containing the origin in its interior, and let P¹ be the algebraic dual polytope of P. Let Q ? Rⁿ×[0,1] be the (n+1)-dimensional polytope that is the convex hull of P×{1} and P¹×{0}. For each face F of P, let Q(F) denote the convex hull of F×{1} and F¹×{0}, where F¹ is the dual face of P¹. Then Q is an antiprism if the set of facets of Q is precisely the collection {Q (F)} for all faces F of P. If Q is an antiprism, the correspondence between primal and dual faces of P and P¹ is manifested in the facets of Q. In this paper, necessary and sufficient conditions for the existence of antiprisms are stated and proved.     

Quotient polytopes of cyclic polytopes part I: Structure and characterization

We investigate the quotient polytopesC/F, whereC is a cyclic polytope andF is a face ofC. We describe the combinatorial structure of such quotients, and show that under suitable restrictions the pair (C, F) is determined by the combinatorial type ofC/F. We describe alternative constructions of these quotients by “splitting vertices” of lower-dimensional cyclic polytopes. Using Gale diagrams, we show that every simpliciald-polytope withd+3 vertices is isomorphic to a quotient of a cyclic polytope.     

Gorenstein toric Fano varieties

We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalizations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.Mathematics Subject Classification (2000): 14J45, 14M25, 52B20Acknowledgement The author would like to thank his thesis advisor Professor Victor Batyrev for posing problems, his advice and encouragement, as well as Professor Günter Ewald for giving reference to [Wir97] and Professor Klaus Altmann for the possibility of giving a talk at the FU Berlin. The author would also like to thank Professor Maximilian Kreuzer for the support with the computer package PALP, the classification data and many examples. Finally the author is grateful to the anonymous referee for corrections and many useful suggestions. The author was supported by DFG, Forschungsschwerpunkt Globale Methoden in der komplexen Geometrie. This work is part of the authors thesis.     

Extensions of regular polytopes with preassigned Schläfli symbol

We say that a (d+1)-polytope P is an extension of a polytope K if the facets or the vertex figures of P are isomorphic to K. The Schläfli symbol of any regular extension of a regular polytope is determined except for its first or last entry. For any regular polytope K we construct regular extensions with any even number as first entry of the Schläfli symbol. These extensions are lattices if K is a lattice. Moreover, using the so-called CPR graphs we provide a more general way of constructing extensions of polytopes.     

Stringy E-functions of varieties with A-D-E singularities

The stringy E-function for normal irreducible complex varieties with at worst log terminal singularities was introduced by Batyrev. It is defined by data from a log resolution. If the variety is projective and Gorenstein and the stringy E-function is a polynomial, Batyrev also defined the stringy Hodge numbers as a generalization of the Hodge numbers of nonsingular projective varieties, and conjectured that they are nonnegative. We compute explicit formulae for the contribution of an A-D-E singularity to the stringy E-function in arbitrary dimension. With these results we can say when the stringy E-function of a variety with such singularities is a polynomial and in that case we prove that the stringy Hodge numbers are nonnegative. Research Assistant of the Fund for Scientific Research - Flanders (Belgium) (F.W.O.),     

Sperner labellings: A combinatorial approach

In 2002, De Loera, Peterson and Su proved the following conjecture of Atanassov: let T be a triangulation of a d-dimensional polytope P with n vertices v₁,v₂,…,vn; label the vertices of T by 1,2,…,n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if vj is on F; then there are at least n−d simplices labelled with d+1 different labels. We prove a generalisation of this theorem which refines this lower bound and which is valid for a larger class of objects.     

Extended formulations for matroid polytopes through randomized protocols

The hitting number of a polytope P is the smallest size of a subset of vertices of P such that every facet of P has a vertex in the subset. We show that, if P is the base polytope of any matroid, then P admits an extended formulation of linear size on the hitting number of P. Our results generalize those of the spanning tree polytope given by Martin and Wong, and extend to polymatroids.     

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.     

On the facets of the secondary polytope

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very well studied, there is not much known about the facets of the secondary polytope.The splits of a polytope, subdivisions with exactly two maximal faces, are the simplest examples of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of these studies and as a starting point of an examination of the subdivisions corresponding to the facets of the secondary polytope in general. As a special case, the notion of k-split is introduced as a possibility to classify polytopes in accordance to the complexity of the facets of their secondary polytopes. An application to matroid subdivisions of hypersimplices and tropical geometry is given.     

On separation and adjacency problems for perfectly matchable subgraph polytopes of a graph

A subgraph F of graph G is called a perfectly matchable subgraph if F contains a set of independent edges convering all the vertices in F. The convex hull of the incidence vectors of perfectly matchable subgraphs of G is a 0–1 polytope. We characterize the adjacency of vertices on such polytopes. We also show that when G is bipartite, the separation problem for such polytones can be solved by maximum flow algorithms.     

Ideal polytopes and face structures of some combinatorial optimization problems

Given a finite setX and a family of feasible subsetsF ofX, the 0–1 polytope P (F is defined as the convex hull of all the characteristic vectors of members ofF We show that under a certain assumption a special type of face ofP(F) is equivalent to the ideal polytope of some pseudo-ordered set. Examples of families satisfying the assumption are those related to the maximum stable set problem, set packing and set partitioning problems, and vertex coloring problem. Using this fact, we propose a new heuristic for such problems and give results of our preliminary computational experiments for the maximum stable set problem.Supported by a JSPS Fellowship for Young Scientists.Supported by Grant-in-Aids for Co-operative Research (06740147) of the Ministry of Education, Science and Culture.     

The Reflexive Dimension of a Lattice Polytope

The reflexive dimension refldim(P) of a lattice polytope P is the minimal integer d so that P is the face of some d-dimensional reflexive polytope. We show that refldim(P) is finite for every P, and give bounds for refldim(kP) in terms of refldim(P) and k. Received July 2, 2004     

An upper bound for the diameter of a polytope

The distance between two vertices of a polytope is the minimum number of edges in a path joining them. The diameter of a polytope is the greatest distance between two vertices of the polytope. We show that if P is a d-dimensional polytope with n facets, then the diameter of P is at most $\text{1}\text{3}\text{2}{}^{\mathrm{d?3}}\text{(n?d+}\text{5}\text{2}\text{)}$.     

Newton Polytopes and Witness Sets

We present two algorithms that compute the Newton polytope of a polynomial f defining a hypersurface \({\mathcal{H}}\) in \({\mathbb{C}^n}\) using numerical computation. The first algorithm assumes that we may only compute values of f—this may occur if f is given as a straight-line program, as a determinant, or as an oracle. The second algorithm assumes that \({\mathcal{H}}\) is represented numerically via a witness set. That is, it computes the Newton polytope of \({\mathcal{H}}\) using only the ability to compute numerical representatives of its intersections with lines. Such witness set representations are readily obtained when \({\mathcal{H}}\) is the image of a map or is a discriminant. We use the second algorithm to compute a face of the Newton polytope of the Lüroth invariant, as well as its restriction to that face.     

Computation of Weng's rank 2 zeta function over an algebraic number field

In this paper, we study the zeta function, named non-abelian zeta function, defined by Lin Weng. We can represent Weng's rank r zeta function of an algebraic number field F as the integration of the Eisenstein series over the moduli space of the semi-stable OF-lattices with rank r. For r=2, in the case of F=Q, Weng proved that it can be written by the Riemann zeta function, and Lagarias and Suzuki proved that it satisfies the Riemann hypothesis. These results were generalized by the author to imaginary quadratic fields and by Lin Weng to general number fields. This paper presents proofs of both these results. It derives a formula (first found by Weng) for Weng's rank 2 zeta functions for general number fields, and then proves the Riemann hypothesis holds for such zeta functions.     

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.