Related papers: Ehrhart polynomials of partial permutohedra

Hibi showed that the polynomial in the numerator of the Ehrhart series of a reflexive polytope is palindromic. We proved that those in the numerator of the Ehrhart series of every graph polytope (defined later) of the bipartite graph is…

In the last decade, the order polytope of the zigzag poset has been thoroughly studied. A related poset, called \emph{crown poset}, obtained by adding an extra relation between the endpoints of an even zigzag poset, is not so well…

Let A and M be nonempty sets of positive integers. A partition of the positive integer n with parts in A and multiplicities in M is a representation of n in the form n = \sum_{a\in A} m_a a, where m_a is in M U {0} for all a in A, and m_a…

For an integral convex polytope $\mathcal{P} \subset \mathbb{R}^d$, we recall $L_\mathcal{P}(n)=|n\mathcal{P} \cap \mathbb{Z}^d|$ the Ehrhart polynomial of $\mathcal{P}$. Let $g_r(\mathcal{P})$ be the $r$th coefficients of…

This paper gives an explicit formula for the Ehrhart quasi-polynomial of certain 2-dimensional polyhedra in terms of invariants of surface quotient singularities. Also, a formula for the dimension of the space of quasi-homogeneous…

We investigate the Ehrhart polynomial for the class of 0-symmetric convex lattice polytopes in Euclidean $n$-space $\mathbb{R}^n$. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima are closely related…

The Ehrhart quasi-polynomial of a rational polytope $P$ is a fundamental invariant counting lattice points in integer dilates of $P$. The quasi-period of this quasi-polynomial divides the denominator of $P$ but is not always equal to it:…

Let $M_{k,m}$ be the space of Laurent polynomials in one variable $x^k + t_1 x^{k-1}+... t_{k+m}x^{-m},$ where $k,m\geq 1$ are fixed integers and $t_{k+m}\neq 0$. According to B. Dubrovin \cite{D}, $M_{k,m}$ can be equipped with a…

We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of a…

We call the $\delta$-vector of an integral convex polytope of dimension $d$ flat if the $\delta$-vector is of the form $(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)$, where $a \geq 1$. In this paper, we give the complete characterization of…

We give explicit polynomial-sized (in $n$ and $k$) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree $k$ in $n$ variables. These convex cones form a family of…

We consider the Ehrhart polynomial of hypersimplices. It is proved that these polynomials have positive coefficients and we give a combinatorial formula for each of them. This settles a problem posed by Stanley and also proves that uniform…

The set ${\cal P}^{m\times n}_{r,d}$ of $m \times n$ complex matrix polynomials of grade $d$ and (normal) rank at most $r$ in a complex $(d+1)mn$ dimensional space is studied. For $r = 1, \dots , \min \{m, n\}-1$, we show that ${\cal…

A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry.…

Let $P_1,\dots,P_k \colon {\bf Z} \to {\bf Z}$ be polynomials of degree at most $d$ for some $d \geq 1$, with the degree $d$ coefficients all distinct, and admissible in the sense that for every prime $p$, there exists integers $n,m$ such…

We show that every (possibly unbounded) convex polygon $P$ in $R^2$ with $m$ edges can be represented by inequalities $p_1 \ge 0,...,p_n \ge 0,$ where the $p_i$'s are products of at most $k$ affine functions each vanishing on an edge of $P$…

We present a short proof of S. Parsa's theorem that there exists a compact $n$-polyhedron $P$, $n\ge 2$, non-embeddable in $\mathbb R^{2n}$, such that $P*P$ embeds in $\mathbb R^{4n+2}$. This proof can serve as a showcase for the use of…

Let $\rho$ be a metric on the set $X=\{1,2,\dots,n+1\}$. Consider the $n$-dimensional polytope of functions $f:X\rightarrow \mathbb{R}$, which satisfy the conditions $f(n+1)=0$, $|f(x)-f(y)|\leq \rho(x,y)$. The question on classifying…

It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…

Consider a finite collection of affine hyperplanes in $\mathbb R^d$. The hyperplanes dissect $\mathbb R^d$ into finitely many polyhedral chambers. For a point $x\in \mathbb R^d$ and a chamber $P$ the metric projection of $x$ onto $P$ is the…