Lattice Polytopes Research Articles

Families of lattice polytopes of mixed degree one

It has been shown by Soprunov that the normalized mixed volume (minus one) of an n-tuple of n-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined n-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples. We give a finiteness result regarding this problem in general dimension n≥4, showing that all but finitely many n-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex Δn−1. Furthermore, we give a complete solution in dimension n=3. In the course of this we show that our finiteness result does not extend to dimension n=3, as we describe infinite families of triples of mixed degree one not admitting a common lattice projection onto the unimodular triangle Δ2.

Correction to: Distance between vertices of lattice polytopes

The erratum mostly concerns Table 4 and Figure 6 where two polytopes were misrepresented in the original version.

Weighted Lattice Point Sums in Lattice Polytopes, Unifying Dehn–Sommerville and Ehrhart–Macdonald

Let V be a real vector space of dimension n and let $$M\subset V$$ be a lattice. Let $$P\subset V$$ be an n-dimensional polytope with vertices in M, and let $$\varphi :V\rightarrow {\mathbb {C}}$$ be a homogeneous polynomial function of degree d. For $$q\in {\mathbb {Z}}_{>0}$$ and any face F of P, let $$D_{\varphi ,F} (q)$$ be the sum of $$\varphi $$ over the lattice points in the dilate qF. We define a generating function $$G_{\varphi }(q,y) \in {\mathbb {Q}}[q] [y]$$ packaging together the various $$D_{\varphi ,F} (q)$$ , and show that it satisfies a functional equation that simultaneously generalizes Ehrhart–Macdonald reciprocity and the Dehn–Sommerville relations. When P is a simple lattice polytope (i.e., each vertex meets n edges), we show how $$G_{\varphi }$$ can be computed using an analogue of Brion–Vergne’s Euler–Maclaurin summation formula.

Simplicial complexes and tilting theory for Brauer tree algebras

We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the lattice polytope corresponding to the simplicial complex of 2-term tilting complexes. Via a geometric interpretation of derived equivalences, we show that the f-vector of the simplicial complexes of Brauer tree algebras only depends on the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support τ-tilting modules, is a derived invariant. Moreover, we apply our result to the enumeration problem of Coxeter-biCatalan combinatorics.

Elementary moves on lattice polytopes

We introduce a graph structure on the set of Euclidean polytopes. The vertices of this graph are the d-dimensional polytopes contained in Rd and its edges connect any two polytopes that can be obtained from one another by either inserting or deleting a vertex, while keeping their vertex sets otherwise unaffected. We prove several results on the connectivity of this graph, and on a number of its subgraphs. We are especially interested in several families of subgraphs induced by lattice polytopes, such as the subgraphs induced by the lattice polytopes with n or n+1 vertices, that turn out to exhibit intriguing properties.

A Continuous Family of Marked Poset Polytopes

For any marked poset we define a continuous family of polytopes, parametrized by a hypercube, generalizing the notions of marked order and marked chain polytopes. By providing transfer maps, we show that the vertices of the hypercube parametrize an Ehrhart equivalent family of lattice polytopes. The combinatorial type of the polytopes is constant when the parameters vary in the relative interior of each face of the hypercube. Moreover, with the help of a subdivision arising from a tropical hyperplane arrangement associated to the marked poset, we give an explicit description of the vertices of the polytope for generic parameters.

Polyhedral realizations of crystal bases and convex-geometric Demazure operators

The main object in this paper is a certain rational convex polytope whose lattice points give a polyhedral realization of a highest weight crystal basis. This is also identical to a Newton-Okounkov body of a flag variety, and it gives a toric degeneration. In this paper, we prove that a specific class of this polytope is given by Kiritchenko's Demazure operators on polytopes. This implies that polytopes in this class are all lattice polytopes. As an application, we give a sufficient condition for the corresponding toric variety to be Gorenstein Fano.

Computational determination of the largest lattice polytope diameter

A lattice (d,k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let δ(d,k) be the largest diameter over all lattice (d,k)-polytopes. We develop a computational framework to determine δ(d,k) for small instances. We show that δ(3,4)=7 and δ(3,5)=9; that is, we verify for (d,k)=(3,4) and (3,5) the conjecture whereby δ(d,k) is at most ⌊(k+1)d∕2⌋ and is achieved, up to translation, by a Minkowski sum of lattice vectors.

Local optimality of Zaks-Perles-Wills simplices

In 1982, Zaks, Perles and Wills discovered a d-dimensional lattice simplex Sd,k with k interior lattice points, whose volume is linear in k and doubly exponential in the dimension d. It is conjectured that, for all d≥3 and k≥1, the simplex Sd,k is a volume maximizer in the family Pd(k) of all d-dimensional lattice polytopes with k interior lattice points. The simplex Sd,k has a facet with only one lattice point in the relative interior. We show that Sd,k is a volume maximizer in the family of simplices S∈Sd(k) that have a facet with one lattice point in its relative interior. We also show that, in the above family, the volume maximizer is unique up to unimodular transformations.

Hollow polytopes of large width

We construct the first known hollow lattice polytopes of width larger than dimension: a hollow lattice polytope (resp., a hollow lattice simplex) of dimension 14 14 (resp., 404 404 ) and of width 15 15 (resp., 408 408 ). We also construct a hollow (nonlattice) tetrahedron of width 2 + 2 2+\sqrt 2 , and we conjecture that this is the maximum width among 3 3 -dimensional hollow convex bodies. We show that the maximum lattice width grows (at least) additively with d d . In particular, the constructions above imply the existence of hollow lattice polytopes (resp., hollow simplices) of arbitrarily large dimension d d and width ≃ 1.14 d \simeq 1.14 d (resp., ≃ 1.01 d \simeq 1.01 d ).

Gorenstein simplices with a given [formula omitted]-polynomial

To classify the lattice polytopes with a given δ-polynomial is an important open problem in Ehrhart theory. A complete classification of the Gorenstein simplices whose normalized volumes are prime integers is known. In particular, their δ-polynomials are of the form 1+tk+⋯+t(v−1)k, where k and v are positive integers. In the present paper, a complete classification of the Gorenstein simplices with the above δ-polynomials will be performed, when v is either p2 or pq, where p and q are prime integers with p≠q. Moreover, we consider the number of Gorenstein simplices, up to unimodular equivalence, with the expected δ-polynomial.

Stringy -functions of canonical toric Fano threefolds and their applications

Let be a -dimensional lattice polytope containing exactly one interior lattice point. We give a simple combinatorial formula for computing the stringy -function of the -dimensional canonical toric Fano variety associated with . Using the stringy Libgober– Wood identity and our formula, we generalize the well-known combinatorial identity which holds for -dimensional reflexive polytopes .

Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series

The Hilbert series of a polarized algebraic variety (X, D) is a powerful invariant that, while it captures some features of the geometry of (X, D) precisely, often cannot recover much information about its singular locus. This work explores the extent to which the Hilbert series of an orbifold del Pezzo surface fails to pin down its singular locus, which provides nonexistence results describing when there are no orbifold del Pezzo surfaces with a given Hilbert series, supplies bounds on the number of singularities on such surfaces, and has applications to the combinatorics of lattice polytopes in the toric case.

Local formulas for Ehrhart coefficients from lattice tiles

As shown by McMullen in 1983, the coefficients of the Ehrhart polynomial of a lattice polytope can be written as a weighted sum of facial volumes. The weights in such a local formula depend only on the outer normal cones of faces, but are far from being unique. In this paper, we develop an infinite class of such local formulas. These are based on choices of fundamental domains in sublattices and obtained by polyhedral volume computations. We hereby also give a kind of geometric interpretation for the Ehrhart coefficients. Since our construction gives us a great variety of possible local formulas, these can, for instance, be chosen to fit well with a given polyhedral symmetry group. In contrast to other constructions of local formulas, ours does not rely on triangulations of rational cones into simplicial or even unimodular ones.

On the Relationship Between Ehrhart Unimodality and Ehrhart Positivity

For a given lattice polytope, two fundamental problems within the field of Ehrhart theory are (1) to determine if its (Ehrhart) $$h^*$$ -polynomial is unimodal and (2) to determine if its Ehrhart polynomial has only positive coefficients. The former property of a lattice polytope is known as Ehrhart unimodality and the latter property is known as Ehrhart positivity. These two properties are often simultaneously conjectured to hold for interesting families of lattice polytopes, yet they are typically studied in parallel. As to answer a question posed at the 2017 Introductory Workshop to the MSRI Semester on Geometric and Topological Combinatorics, the purpose of this note is to show that there is no general implication between these two properties in any dimension greater than two. To do so, we investigate these two properties for families of well-studied lattice polytopes, assessing one property where previously only the other had been considered. Consequently, new examples of each phenomena are developed, some of which provide an answer to an open problem in the literature. The well-studied families of lattice polytopes considered include zonotopes, matroid polytopes, simplices of weighted projective spaces, empty lattice simplices, smooth polytopes, and $${\varvec{s}}$$ -lecture hall simplices.

Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP

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.

Ehrhart Polynomial Roots of Reflexive Polytopes

Recent work has focused on the roots $z\in\mathbb{C}$ of the Ehrhart polynomial of a lattice polytope $P$. The case when $\Re{z}=-1/2$ is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when $\mathrm{dim}(P)\leq 7$. We also consider the "half-strip condition", where all roots $z$ satisfy $-\mathrm{dim}(P)/2\leq\Re{z}\leq \mathrm{dim}(P)/2-1$, and show that this holds for any reflexive polytope with $\mathrm{dim}(P)\leq 5$. We give an example of a $10$-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi–Shibata in dimension $34$.

H*-Polynomials with Roots on the Unit Circle

For an n-dimensional lattice simplex with vertices given by the standard basis vectors and where has positive entries, we investigate when the Ehrhart -polynomial for factors as a product of geometric series in powers of z. Our motivation is a theorem of Rodriguez-Villegas implying that when the -polynomial of a lattice polytope P has all roots on the unit circle, then the Ehrhart polynomial of P has positive coefficients. We focus on those for which has only two or three distinct entries, providing both theoretical results and conjectures/questions motivated by experimental evidence.

Interlacing polynomials and the veronese construction for rational formal power series

AbstractFixing a positive integer r and $0 \les k \les r-1$, define $f^{\langle r,k \rangle }$ for every formal power series f as $ f(x) = f^{\langle r,0 \rangle }(x^r)+xf^{\langle r,1 \rangle }(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle }(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\, h(x) := ( (1+x+\cdots +x^{r-1})^{n} h(x) )^{\langle r,k \rangle }$ has only non-positive zeros for any $r \ges \deg h(x) -k$ and any positive integer n. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension n, which states that $U^{n}_{r,0}\,h(x)$ has only negative, real zeros whenever $r\ges n$. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence $( h^{\langle r,r-i \rangle }(x))_{1\les i \les r}$ is interlacing, so is $( U^{n}_{r,r-i}\, h(x) )_{1\les i \les r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for coloured permutations. Besides, we derive a Carlitz identity for refined coloured permutations.

Regular switching components

We consider a problem of Discrete Tomography which consists of reconstructing a finite lattice set S⊂Z2 from given horizontal and vertical X-rays (i.e. with prescribed numbers of points in each row and column). Without additional requirements, the problem can be solved in polynomial time. Many variants require the solution to be in a chosen class A. For instance, the problem is NP-complete for the class A=H∩V of HV-convex lattice sets and it is polynomial for the class A=H∩V∩P of HV-convex polyominoes. Twenty years after these results, the problem's complexity remains unknown for A=C, the class of C-convex lattice sets (i.e. two-dimensional lattice polytopes).The main difficulty to solve this problem comes from combinatorial structures called switching components. Switching components are closed path with horizontal and vertical edges such that the solutions S are composed of either the elements of even or of odd indices. This binary choice can be encoded by a Boolean variable associated with the switching component and the convexity constraints are simply encoded by SAT clauses (2-clauses for HV-convexity and 3-clauses for C-convexity).The purpose of the paper is to investigate the properties of the switching components and the consequences of the convexity requirements. We divide the switching components in two classes: regular if their turning angle is constant, irregular otherwise. We prove that adjacent regular switching components have the same Boolean values. This property allows us to merge them into extended switching components. We prove that if all switching components are regular, then the extended switching components are all independent (then the number of solutions with the considered feet is 2n, where n is the number of extended switching components). Finally, we prove that they are geometrically ordered.