Related papers: The polytope of Tesler matrices
We define the $k$-dimensional generalized Euler function $\varphi_k(n)$ as the number of ordered $k$-tuples $(a_1,\ldots,a_k)\in {\Bbb N}^k$ such that $1\le a_1,\ldots,a_k\le n$ and both the product $a_1\cdots a_k$ and the sum $a_1+\cdots…
The aim of this paper is to study alcoved polytopes, which are polytopes arising from affine Coxeter arrangements. This class of convex polytopes includes many classical polytopes, for example, the hypersimplices. We compare two…
For systems of polynomial equations, we study the problem of computing the Newton polytope of their eliminants. As was shown by Esterov and Khovanskii, such Newton polytopes are mixed fiber polytopes of the Newton polytopes of the input…
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…
Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the "moments" F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a…
We describe a construction for d-polytopes generalising the well known stacking operation. The construction is applied to produce 2-simplicial and 2-simple 4-polytopes with g_2=0 on any number of n >= 13 vertices. In particular, this…
We expand on a result of Barvinok and Hartigan to derive asymptotic formulas for the number of integer and binary integer points in a wide class of multi-index $k_1\times k_2\times \ldots \times k_{\nu}$ transportation polytopes. A simple…
In this paper we consider polytopes given by systems of $n$ inequalities in $d$ variables, where every inequality has at most two variables with nonzero coefficient. We denote this family by $LI(2)$. We show that despite of the easy…
In this paper we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb{R}^n$, where $e_1,\dots,e_n$ is the standard basis of…
The random beta polytope is defined as the convex hull of $n$ independent random points with the density proportional to $(1-\|x\|^2)^\beta$ on the $d$-dimensional unit ball, where $\beta>-1$ is a parameter. Similarly, the random beta'…
Given a positive integer $n$ and a partition $(n_1,\ldots,n_r)$ of $n$, one can consider the associated $n$-dimensional multiprojective space $\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_r}$. These multiprojective spaces are…
The main aim of this paper is to provide a method which allows finding limiting shapes of symbolic generic initial systems of higher-dimensional subvarieties of P^n. M. Mustata and S. Mayes established a connection between volumes of…
It is well-known that a quiver Q of type A_n is representation-finite, and that its indecomposable representations are thin (all Jordan-Hoelder multiplicities are 0 or 1). By now, various methods of proof are known. The aim of this note is…
We consider the set $\mathcal M_n\left(\mathbb Z; H\right)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain and asymptotic formula on the number of matrices from $\mathcal M_n\left(\mathbb Z; H\right)$ with…
The convex hull $P_{n}$ of a Gaussian sample $X_{1},...,X_{n}$ in $R^{d}$ is a Gaussian polytope. We prove that the expected number of facets $E f_{d-1} (P_n)$ is monotonically increasing in $n$. Furthermore we prove this for random…
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 describe the computation of polytope volumes by descent in the face lattice, its implementation in Normaliz, and the connection to reverse-lexicographic triangulations. The efficiency of the algorithm is demonstrated by several high…
Asymptotic tensor rank is notoriously difficult to determine. Indeed, determining its value for the $2\times 2$ matrix multiplication tensor would determine the matrix multiplication exponent, a long-standing open problem. On the other…
We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant…
In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the non-empty subsets $A\subset \{1,..,n\}$, is it possible to construct a body $T\subset \mathbb{R}^n$ such that $x_A=|T_A|$ where $|T_A|$…