We prove a sharp upper bound on the number of distinct columns of a totally unimodular matrix with column sums 1 improving upon Heller's classical bound. The proof uses Seymour's decomposition theorem. Such matrices are closely related to unimodular polytopes: lattice polytopes where the vertices of every full-dimensional subsimplex form an affine lattice basis. This is an interesting subclass of 0/1-polytopes and contains for instance edge polytopes of bipartite graphs. Our main result on totally unimodular matrices implies a sharp upper bound on the number of vertices of unimodular polytopes.

In this note we prove that in fixed dimension the Ehrhart h^*-polynomial of a lattice polytope of sufficiently large lattice width is real-rooted. In particular, this implies strict log-concavity and unimodality of the h^*-vector and answers a question of Averkov, Hofscheier and the author. For a lattice simplex we prove the analogous statement for its local h^*-polynomial, also called box polynomial. The proofs were found using ChatGPT 5.6 Sol and follow essentially directly from a result by Basu and Oertel that for large enough lattice width counting lattice points approximates the volume.

Lattice polytopes are called very ample if for every sufficiently large k every lattice point of height k in the cone over the lattice polytope is the sum of k lattice points of height 1. This is a weakening of the well-known integer decomposition property (also called IDP). We give an example of a very ample lattice polytope whose h^*-vector is non-unimodal. Here, the h^*-vector is the coefficient vector of the numerator of the Ehrhart series of the lattice polytope. This answers a question of Ferroni and Higashitani, as well as a related question by Balletti. The main question whether IDP lattice polytopes have unimodal h^*-vector is still open. The example was found using ChatGPT 5.6 Sol. It is just the Cartesian square of a lattice polytope belonging to a class of very ample examples constructed by Lasoń and Michalek.

Given lattice polytopes P_1, …, P_k contained in a k-dimensional subspace U ⊆ℝ^d and a d-dimensional lattice polytope Q ⊂ℝ^d, we compute the Hodge vector of the Cayley polytope P_1 * ⋯ * P_k * Q, and show that it equals the mixed volume of P_1, …, P_k times the Hodge vector of the projection of Q along U. Here, the Hodge vector of a lattice polytope is its local h^*-vector with leading and trailing zeroes removed. This result allows finding infinitely many high-dimensional lattice polytopes with the same Hodge vector that are not free joins. The proof relies on a closed formula for the Hodge-Deligne polynomial of generic complete intersections in the torus in terms of the bivariate/mixed h^*-polynomial. A special case of our construction is what we call Lawrence twists: extending the Gale transform by centrally-symmetric pairs of vectors. As applications, we can produce many new thin polytopes answering a question by Borger, Kretschmer and the second author, and we provide an alternative explanation of the thinness of B_k-polytopes answering a question of Selyanin.

Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a kth dilation is a sum of k lattice points in the polytope. It is a long-standing conjecture whether the numerator of the Ehrhart series of an IDP polytope, called the h^*-polynomial, has a unimodal coefficient vector. In this preliminary report on research in progress we present examples showing that h^*-vectors of IDP polytopes do not have to be log-concave. This answers a question of Luis Ferroni and Akihiro Higashitani. As this is an ongoing project, this paper will be updated with more details and examples in the near future.

An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension: ◦ We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension $10$ and volume up to $2^{31}$ . Among them, we find five empty ones of width $11$ and none of larger width. ◦ Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension d and width growing asymptotically as $d/\operatorname {\mathrm {arcsinh}}(1) \sim 1.1346\,d$ .

In this paper we study the novel notion of thin polytopes: lattice polytopes whose local $h^*$-polynomials vanish. The local $h^*$-polynomial is an important invariant in modern Ehrhart theory. Its definition goes back to Stanley with fundamental results achieved by Karu, Borisov & Mavlyutov, Schepers, and Katz & Stapledon. The study of thin simplices was originally proposed by Gelfand, Kapranov and Zelevinsky, where in this case the local $h^*$-polynomial simply equals its so-called box polynomial. Our main results are the complete classification of thin polytopes up to dimension 3 and the characterization of thinness for Gorenstein polytopes. The paper also includes an introduction to the local $h^*$-polynomial with a survey of previous results.

A d-dimensional lattice polytope P is Gorenstein if it has a multiple r P that is a reflexive polytope up to translation by a lattice vector. The difference d + 1 - r is called the degree of P. We show that a Gorenstein polytope is a lattice pyramid if its dimension is at least three times its degree. This was previously conjectured by Batyrev and Juny. We also present a refined conjecture and prove it for IDP Gorenstein polytopes.

We give an upper bound on the volume vol(P*) of a polytope P* dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree (-K_X)^d of a d-dimensional toric Fano variety X with at worst canonical singularities.

We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. Following an argument of Eisenbrand and Shmonin, we prove that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators (and the lattice width of their convex hull) is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.

We give an explicit upper bound on the volume of lattice simplices with fixed positive number of interior lattice points. The bound differs from the conjectural sharp upper bound only by a linear factor in the dimension. This improves significantly upon the previously best results by Pikhurko from 2001.

The mixed discriminant of a family of point configurations can be considered as a generalization of the A A -discriminant of one Laurent polynomial to a family of Laurent polynomials. Generalizing the concept of defectivity, a family of point configurations is called defective if the mixed discriminant is trivial. Using a recent criterion by Furukawa and Ito we give a necessary condition for defectivity of a family in the case that all point configurations are full-dimensional. This implies the conjecture by Cattani, Cueto, Dickenstein, Di Rocco, and Sturmfels that a family of n n full-dimensional configurations in Z n {\mathbb {Z}}^n is defective if and only if the mixed volume of the convex hulls of its elements is 1 1 .

The characterization of lattice polytopes based upon information about their Ehrhart h^* -polynomials is a difficult open problem. In this paper, we finish the classification of lattice polytopes whose h^* -polynomials satisfy two properties: they are palindromic (so the polytope is Gorenstein) and they consist of precisely three terms. This extends the classification of Gorenstein polytopes of degree two due to Batyrev and Juny. The proof relies on the recent characterization of Batyrev and Hofscheier of empty lattice simplices whose h^* -polynomials have precisely two terms. Putting our theorem in perspective, we give a summary of these and other existing results in this area.

The degree of a lattice polytope is a notion in Ehrhart theory that was studied quite intensively over the previous years. It is well-known that a lattice polytope has normalized volume one if and only if its degree is zero. Recently, Esterov and Gusev gave a complete classification result of families of $n$ lattice polytopes in $\mathbb{R}^n$ whose mixed volume equals one. Here, we give a reformulation of their result involving the novel notion of a mixed degree that generalizes the degree similar to how the mixed volume generalizes the volume. We discuss and motivate this terminology, and explain why it extends a previous definition of Soprunov. We also remark how a recent combinatorial result due to Bihan solves a related problem posed by Soprunov.

Generalizing the famous Bernstein-Kushnirenko Theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskii in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.

A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. We show as a corollary to a general result in the Ehrhart theory of lattice polytopes that the $h^*$-vector of a spanning lattice polytope has no gaps, i. e., $h^*_i =0$ implies $h^*_{i+1}=0$. This generalizes a recent result by Blekherman, Smith, and Velasco, and implies a polyhedral consequence of the Eisenbud-Goto conjecture. We also discuss how this relates to unimodality questions of lattice polytopes and previously achieved decomposition results on lattice polytopes of given degree.

We present examples of smooth lattice polytopes in dimensions 3 and higher with the maximal possible number of negative Ehrhart coefficients. This answers a question by Bruns. We also discuss Berline–Vergne valuations as a useful tool in proving Ehrhart positivity results.

The Ehrhart polynomial of a lattice polygon P is completely determined by the pair (b(P), i(P)) where b(P) equals the number of lattice points on the boundary and i(P) equals the number of interior lattice points. All possible pairs (b(P), i(P)) are completely described by a theorem due to Scott. In this note, we describe the shape of the set of pairs (b(T), i(T)) for lattice triangles T by finding infinitely many new Scott-type inequalities.