We give an effective upper bound on the h^*-polynomial of a lattice polytope in terms of its degree and leading coefficient, confirming a conjecture of Batyrev. We deduce this bound as a consequence of a strong Cayley decomposition theorem which says, roughly speaking, that any lattice polytope with a large multiple that has no interior lattice points has a nontrivial decomposition as a Cayley sum of polytopes of smaller dimension. In an appendix, we interpret this result in terms of adjunction theory for toric varieties.
In this paper, we give an algebro-geometric characterization of Cayley polytopes. As a special case, we also characterize lattice polytopes with lattice width one by using Seshadri constants.
Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun's conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to two one-variable deformations of Cayley polytopes (which we call t-Cayley and t-Gayley polytopes), and to the most general two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph. Our approach is based on an explicit triangulation …
We define Q-normal lattice polytopes. Natural examples of such polytopes are Cayley sums of strictly combinatorially equivalent lattice polytopes, which correspond to particularly nice toric fibrations, namely toric projective bundles. In a recent paper Batyrev and Nill have suggested that there should be a bound, N(d), such that every lattice polytope of degree d and dimension at least N(d) decomposes as a Cayley sum. We give a sharp answer to this question for smooth Q-normal polytopes. We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a Cayley sum of strictl…
A theorem of Howe states that every 3-dimensional lattice polytope $P$ whose only lattice points are its vertices, is a Cayley polytope, i.e. $P$ is the convex hull of two lattice polygons with distance one. We want to generalize this result by classifying 3-dimensional lattice polytopes without interior lattice points. The main result will be, that they are up to finite many exceptions either Cayley polytopes or there is a projection, which maps the polytope to the double unimodular 2-simplex. To every such polytope we associate a smooth projective surface of genus 0.
Lattice polytopes which possess the integer decomposition property (IDP for short) turn up in many fields of mathematics. It is known that if the Cayley sum of lattice polytopes possesses IDP, then so does their Minkowski sum. In this paper, the Cayley sum of the order polytope of a finite poset and the stable set polytope of a finite simple graph is studied. We show that the Cayley sum of an order polytope and the stable set polytope of a perfect graph possesses a regular unimodular triangulation and IDP, and hence so does their Minkowski sum. Moreover, it turns out that, for an order polytop…
We introduce the package LatticePolytopes for Macaulay2. The package provides methods for computations related to Cayley structures, local positivity and smoothness for lattice polytopes.
We provide a complete classification up to isomorphism of all smooth convex lattice 3-polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes and the remaining 4 are obtained as inverse stellar subdivisions of such polytopes. We derive a classification, up to isomorphism, of all smooth embeddings of toric threefolds in $\mathbb{P}^N$ where $N\le 15$. Again we have in total 103 such embeddings. Of these, 99 are projective bundles embedded in $\mathbb{P}^N$ and the remaining 4 are blow-ups of such tor…
Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $r\ge{}2$, and let $n_i$ be the number of vertices of $P_i$, $1\le{}i\le{}r$. It has been shown by Fukuda and Weibel that the number of $k$-faces of the Minkowski sum, $P_1+P_2+...+P_r$, is bounded from above by $Φ_{k+r}(n_1,n_2,...,n_r)$, where $Φ_{\ell}(n_1,n_2,...,n_r)= \sum_{\substack{1\le{}s_i\le{}n_i s_1+...+s_r=\ell}} \prod_{i=1}^r\binom{n_i}{s_i}$, $\ell\ge{}r$. Fukuda and Weibel have also shown that the upper bound mentioned above is tight for $d\ge{}4$, $2\le{}r\le{}\lfloor\frac{d}{2}\rfloor$, and for a…
Cayley's first hyperdeterminant is a straightforward generalization of determinants for tensors. We prove that nonzero hyperdeterminants imply lower bounds on some types of tensor ranks. This result applies to the slice rank introduced by Tao and more generally to partition ranks introduced by Naslund. As an application, we show upper bounds on some generalizations of colored sum-free sets based on constraints related to order polytopes.
We prove that any one-ended, locally finite Cayley graph with non-torsion generators admits a decomposition into edge-disjoint Hamiltonian (i.e. spanning) double-rays. In particular, the $n$-dimensional grid $\mathbb{Z}^n$ admits a decomposition into $n$ edge-disjoint Hamiltonian double-rays for all $n \in \mathbb{N}$.
We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties.
We show that any smooth lattice polytope P with codegree greater or equal than (dim(P)+3)/2 (or equivalently, with degree smaller than dim(P)/2), defines a dual defective projective toric manifold. This implies that P is Q-normal (in the terminology of a recent paper by Di Rocco, Piene and the first author) and answers partially an adjunction-theoretic conjecture by Beltrametti and Sommese. Also, it follows that smooth lattice polytopes with this property are precisely strict Cayley polytopes, which completes the answer of a question of Batyrev and the second author in the nonsingular case.
The paper describes a construction of abstract polytopes from Cayley graphs of symmetric groups. Given any connected graph G with p vertices and q edges, we associate with G a Cayley graph of the symmetric group S_p and then construct a vertex-transitive simple polytope of rank q, called the graphicahedron, whose 1-skeleton (edge graph) is the Cayley graph. The graphicahedron of a graph G is a generalization of the well-known permutahedron; the latter is obtained when the graph is a path. We also discuss symmetry properties of the graphicahedron and determine its structure when G is small.
Let $\mathcal{P} \subset \mathbb{R}^N$ be an integral convex polytope of dimension $d$ and write $k \mathcal{P}$, where $k = 1, 2, \ldots$, for dilations of $\mathcal{P}$. We say that $\mathcal{P}$ possesses the integer decomposition property if, for any integer $k = 1, 2, \ldots$ and for any $α\in k \mathcal{P} \cap \mathbb{Z}^N$, there exist $α_{1}, \ldots, α_k$ belonging to $\mathcal{P} \cap \mathbb{Z}^N$ such that $α= α_1 + \cdots + α_k$. A fundamental question is to determine the integers $k > 0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. I…
A well-known conjecture of Alspach says that every $2k$-regular Cayley graph of an abelian group can be decomposed into Hamiltonian cycles. We consider an analogous question for infinite abelian groups. In this setting one natural analogue of a Hamiltonian cycle is a spanning double-ray. However, a naive generalisation of Alspach's conjecture fails to hold in this setting due to the existence of $2k$-regular Cayley graphs with finite cuts $F$ where $|F|$ and $k$ differ in parity, which necessarily preclude the existence of a decomposition into spanning double-rays. We show that every $4$-regul…
We use a version of localization in equivariant cohomology for the norm-square of the moment map, described by Paradan, to give several weighted decompositions for simple polytopes. As an application, we study Euler-Maclaurin formulas.
The weak-map order on the matroid base polytopes is the partial order defined by inclusion. Lucas proved that the base polytope of no binary matroid includes the base polytope of a connected matroid. A matroid base polytope is said to be decomposable when it has a polytopal decomposition which consists of at least two matroid base polytopes. We shed light on the relation between the decomposability and the weak-map order of matroid base polytopes. We classify matroids into five types with respect to the weak-map order and decomposability. We give an example of a matroid in each class. Moreover…
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial time algorithm for decomposing polygons. For higher dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithm include absolute irreducibility testing and factorization of polynomials via their Newton polytopes.
We explore four approaches to the question of defectivity for a complex projective toric variety $X_A$ associated with an integral configuration $A$. The explicit tropicalization of the dual variety $X_A^\vee$ due to Dickenstein, Feichtner, and Sturmfels allows for the computation of the defect in terms of an affine combinatorial invariant $ρ(A)$. We express $ρ(A)$ in terms of affine invariants $ι(A)$ associated to Esterov's iterated circuits and $λ(A)$, an invariant defined by Curran and Cattani in terms of a Gale dual of $A$. Thus we obtain formulae for the dual defect in terms of iterated c…
Let $P$ be the Gelfand--Tsetlin polytope defined by the skew shape $λ/μ$ and weight $w$. In the case corresponding to a standard Young tableau, we completely characterize for which shapes $λ/μ$ the polytope $P$ is integral. Furthermore, we show that $P$ is a compressed polytope whenever it is integral and corresponds to a standard Young tableau. We conjecture that a similar property hold for arbitrary $w$, namely that $P$ has the integer decomposition property whenever it is integral. Finally, a natural partial ordering on GT-polytopes is introduced that provides information about integrality …
Though much is known about ${\bf s}$-lecture hall polytopes, there are still many unanswered questions. In this paper, we show that ${\bf s}$-lecture hall polytopes satisfy the integer decomposition property (IDP) in the case of monotonic ${\bf s}$-sequences. Given restrictions on a monotonic ${\bf s}$-sequence, we discuss necessary and sufficient conditions for the Fano, reflexive and Gorenstein properties. Additionally, we give a construction for producing Gorenstein/IDP lecture hall polytopes.
This is a continuation of the early paper concerning matroid base polytope decomposition. Here, we will present sufficient conditions on $M$ so its base matroid polytope $P(M)$ has a {\em sequence} of hyperplane splits. The latter yields to decompositions of $P(M)$ with two or more pieces for infinitely many matroids $M$. We also present necessary conditions on the Euclidean representation of rank three matroids $M$ for the existences of decompositions of $P(M)$ into $2$ or $3$ pieces. Finally, we prove that $P(M_1 \oplus M_2)$ has a sequence of hyperplane splits if either $P(M_1)$ or $P(M_2)$…
A $P_\ell$-decomposition of a graph $G$ is a set of paths with $\ell$ edges in $G$ that cover the edge set of $G$. Favaron, Genest, and Kouider (2010) conjectured that every $(2k+1)$-regular graph that contains a perfect matching admits a $P_{2k+1}$-decomposition. They also verified this conjecture for $5$-regular graphs without cycles of length $4$. In 2015, Botler, Mota, and Wakabayashi verified this conjecture for $5$-regular graphs without triangles. In this paper, we verify it for $(2k+1)$-regular graphs that contain the $k$th power of a spanning cycle; and for $5$-regular graphs that con…
Let $P(M)$ be the matroid base polytope of a matroid $M$. A {\em matroid base polytope decomposition} of $P(M)$ is a decomposition of the form $P(M) = \bigcup\limits_{i=1}^t P(M_{i})$ where each $P(M_i)$ is also a matroid base polytope for some matroid $M_i$, and for each $1\le i \neq j\le t$, the intersection $P(M_{i}) \cap P(M_{j})$ is a face of both $P(M_i)$ and $P(M_j)$. In this paper, we investigate {\em hyperplane splits}, that is, polytope decompositions when $t=2$. We give sufficient conditions for $M$ so $P(M)$ has a hyperplane split and characterize when $P(M_1 \oplus M_2)$ has a hyp…
In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric $3$-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.
We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. A sufficient condition for vertex adjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph. We introduce a heuristic general variable neighborhood search algorithm for this problem based on finding a vertex-disjoint cycle cover of the multigraph through reduction to perfect matching and several cycle merging operations. The algorithm has a one-sided error: the answer "not adjacent" is…
Let $\mathcal{P} \subset \mathbb{R}^{d}$ and $\mathcal{Q} \subset \mathbb{R}^e$ be integral convex polytopes of dimension $d$ and $e$ which contain the origin of $\mathbb{R}^{d}$ and $\mathbb{R}^e$, respectively. In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of $\mathcal{P}$ and $\mathcal{Q}$ to possess the integer decomposition property will be presented.
It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency.
Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Gröbner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furt…
The concept of sums of nonnegative circuit polynomials (SONC) was recently introduced as a new certificate of nonnegativity especially for sparse polynomials. In this paper, we explore the relationship between nonnegative polynomials and SONC polynomials. As a first result, we provide sufficient conditions for nonnegative polynomials with general Newton polytopes to be SONC polynomials, which generalizes the previous result on nonnegative polynomials with simplex Newton polytopes. Secondly, we prove that every SONC polynomial admits a SONC decomposition without cancellation. In other words, SO…
Reflexive polytopes form one of the distinguished classes of lattice polytopes. Especially reflexive polytopes which possess the integer decomposition property are of interest. In the present paper, by virtue of the algebraic technique on Grönbner bases, a new class of reflexive polytopes which possess the integer decomposition property and which arise from perfect graphs will be presented. Furthermore, the Ehrhart $δ$-polynomials of these polytopes will be studied.
We use a polynomial decomposition result by Stapledon to show that the numerator polynomial of the Ehrhart series of an open polytope is the difference of two symmetric polynomials with nonnegative integer coefficients. We obtain a related decomposition for order polytopes and for the numerator polynomial of the corresponding series for chromatic polynomials. The nonnegativity of the coefficients in such decompositions provide inequalities satisfied by the coefficients of chromatic polynomials for any simple graph.
In this note we point out the relation between Brion's formula for the lattice point generating function of a convex polytope in terms of the vertex cones [Brion1988] on the one hand, and the polar decomposition à la Lawrence/Varchenko [Lawrence1991, Varchenko1987] on the other. We then go on to prove a version of polar decomposition for non-simple polytopes.
Originally introduced by Fine and Reid in the study of plurigenera of toric hypersurfaces, the Fine interior of a lattice polytope got recently into the focus of research. It is has been used for constructing canonical models in the sense of Mori Theory [arXiv:2008.05814]. Based on the Fine interior, we propose here a modification of the original adjoint polytopes as defined in [arXiv:1105.2415], by defining the Fine adjoint polytope $P^{F(s)}$ of $P$ as consisting of the points in $P$ that have lattice distance at least $s$ to all valid inequalities for $P$. We obtain a Fine Polyhedral Adjunc…
We analyze marked poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appear naturally in the representation theory of semi-simple Lie algebras, and hence we can give a necessary and sufficient condition on the marked poset such that the associated toric degenerations of the corresponding partial flag variety are isomorphic. We further show that the set of lattice points in such a marked poset polytope is the Minkowski sum of sets of lattice points for 0-1 polytopes. Moreov…
The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovász and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a number of indecomposable graphs, so called bricks. For many properties that are of interest on graphs with perfect matchings, including the description of the perfect matching polytope, it suffices to consider these bricks. A key result by Lovász on…
We introduce the notion of a \emph{conic sequence} of a convex polytope. It is a way of building up a polytope starting from a vertex and attaching faces one by one with certain regulations. We apply this to a toric variety to obtain an iterated cofibration structure on it. This allows us to prove several vanishing results in the rational cohomology of a toric variety and to calculate Poincaré polynomials for a large class of singular toric varieties.
We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope into special polyhedra. We use these decompositions and develop methods for computing a special class of integrals exactly. Namely, we are interested in computing the exact value of integrals of polynomial functions over convex polyhedra. We present prior work and new extensions of the integration algorithms. Every integration method we present requires that …
Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove tha…
The complexity of the matching polytope of graphs may be measured with the maximum length $β$ of a starting sequence of odd ears in an ear-decomposition. Indeed, a theorem of Edmonds and Pulleyblank shows that its facets are defined by 2-connected factor-critical graphs, which have an odd ear-decomposition (according to a theorem of Lovász). In particular, $β(G) \leq 1$ if and only if the matching polytope of the graph $G$ is completely described by non-negativity, star and odd-circuit inequalities. This is essentially equivalent to the h-perfection of the line-graph of $G$, as observed by Cao…
