Related papers: Integer points in dilates of polytopes

The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We…

We study integer programming instances over polytopes P(A,b)={x:Ax<=b} where the constraint matrix A is random, i.e., its entries are i.i.d. Gaussian or, more generally, its rows are i.i.d. from a spherically symmetric distribution. The…

We consider the family $\mathrm{MC}_d$ of monic centered polynomials of one complex variable with degree $d \geq 2$, and study the map $\widehat{\Phi}_d:\mathrm{MC}_d\to \widetilde{\Lambda}_d \subset \mathbb{C}^d / \mathfrak{S}_d$ which…

We study integer coefficient polynomials of fixed degree and maximum height $H$, that are irreducible by Dumas's criterion. We call such polynomials Dumas polynomials. We derive upper bounds on the number of Dumas polynomials, as $H$…

For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P…

Let a polytope $P$ be defined by a system $A x \leq b$. We consider the problem of counting the number of integer points inside $P$, assuming that $P$ is $\Delta$-modular, where the polytope $P$ is called $\Delta$-modular if all the rank…

An integer vector $b \in \mathbb{Z}^d$ is a degree sequence if there exists a hypergraph with vertices $\{1,\dots,d\}$ such that each $b_i$ is the number of hyperedges containing $i$. The degree-sequence polytope $\mathscr{Z}^d$ is the…

Let $\mathbf{K}$ be a field and $\phi$, $\mathbf{f} = (f_1, \ldots, f_s)$ in $\mathbf{K}[x_1, \dots, x_n]$ be multivariate polynomials (with $s < n$) invariant under the action of $\mathcal{S}_n$, the group of permutations of $\{1, \dots,…

An integer point in a polyhedron is called irreducible iff it is not the midpoint of two other integer points in the polyhedron. We prove that the number of irreducible integer points in $n$-dimensional polytope with radius $k$ given by a…

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of $n$ real numbers (for short, \emph{grid}). First, we prove that every such grid contains $\Omega(\log n)$ points in convex…

The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n)<=M_{ubt}(d,n) provided by…

In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…

We consider the set $\mathcal{M}_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb Z; H)$ with a given characteristic…

It is generally hard to count, or even estimate, how many integer points lie in a polytope P. Barvinok and Hartigan have approached the problem by way of information theory, showing how to efficiently compute a random vector which samples…

Given a set $S \subseteq \mathbb{R}^d$, a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum number of vertices of hollow polytopes whose facets are…

A wide variety of problems in combinatorics and discrete optimization depend on counting the set $S$ of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour…

We study the number of lattice points in integer dilates of the rational polytope $P = (x_1,...,x_n) \in \R_{\geq 0}^n : \sum_{k=1}^n x_k a_k \leq 1$, where $a_1,...,a_n$ are positive integers. This polytope is closely related to the linear…

Let $f_i(P)$ denote the number of $i$-dimensional faces of a convex polytope $P$. Furthermore, let $S(n,d)$ and $C(n,d)$ denote, respectively, the stacked and the cyclic $d$-dimensional polytopes on $n$ vertices. Our main result is that for…

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review…

In this paper, we investigate two properties concerning the unimodality of the $\delta$-vectors of lattice polytopes, which are log-concavity and alternatingly increasingness. For lattice polytopes $\mathcal{P}$ of dimension $d$, we prove…