Projective toric codes

Any integral convex polytope [Formula: see text] in [Formula: see text] provides an [Formula: see text]-dimensional toric variety [Formula: see text] and an ample divisor [Formula: see text] on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on [Formula: see text], obtained by evaluating global section of the line bundle corresponding to [Formula: see text] on every rational point of [Formula: see text]. This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope [Formula: see text] and an algorithmic technique to get a lower bound on the minimum distance is described.

Integer Decomposition Property of Dilated Polytopes

An integral convex polytope $\mathcal{P} \subset \mathbb{R}^N$ possesses the integer decomposition property if, for any integer $k > 0$ and for any $\alpha \in k \mathcal{P} \cap \mathbb{Z}^{N}$, there exist $\alpha_{1}, \ldots, \alpha_k \in \mathcal{P} \cap \mathbb{Z}^{N}$ such that $\alpha = \alpha_{1} + \cdots + \alpha_k$. A fundamental question is to determine the integers $k > 0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. In the present paper, combinatorial invariants related to the integer decomposition property of dilated polytopes will be proposed and studied.

Non-Normal Very Ample Polytopes and their Holes

In this paper, we show that for given integers $h$ and $d$ with $h \geq 1$ and $d \geq 3$, there exists a non-normal very ample integral convex polytope of dimension $d$ which has exactly $h$ holes.

Classification of Ehrhart polynomials of integral simplices

International audience Let $δ (\mathcal{P} )=(δ _0,δ _1,\ldots,δ _d)$ be the $δ$ -vector of an integral convex polytope $\mathcal{P}$ of dimension $d$. First, by using two well-known inequalities on $δ$ -vectors, we classify the possible $δ$ -vectors with $\sum_{i=0}^d δ _i ≤3$. Moreover, by means of Hermite normal forms of square matrices, we also classify the possible $δ$ -vectors with $\sum_{i=0}^d δ _i = 4$. In addition, for $\sum_{i=0}^d δ _i ≥5$, we characterize the $δ$ -vectors of integral simplices when $\sum_{i=0}^d δ _i$ is prime. Soit $δ (\mathcal{P} )=(δ _0,δ _1,\ldots,δ _d)$ le $δ$ -vecteur d'un polytope intégrante de dimension d. Tout d'abord, en utilisant deux bien connus des inégalités sur $δ$ -vecteurs, nous classons les $δ$ -vecteurs possibles avec $\sum_{i=0}^d δ _i ≤3$ En outre, par le biais de Hermite formes normales, nous avons également classer les $δ$ -vecteurs avec $\sum_{i=0}^d δ _i = 4$. De plus, pour $\sum_{i=0}^d δ _i ≥5$, nous caractérisons les $δ$-vecteurs des simplex inégalités lorsque $\sum_{i=0}^d δ _i$ est premier.

Higher Spin Alternating Sign Matrices

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer $r$, all complete row and column sums are $r$, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin $r/2$ statistical mechanical vertex models with domain-wall boundary conditions. The case $r=1$ gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size $n$ are the integer points of the $r$-th dilate of an integral convex polytope of dimension $(n{-}1)^2$ whose vertices are the standard alternating sign matrices of size $n$. It then follows that, for fixed $n$, these matrices are enumerated by an Ehrhart polynomial in $r$.

The $h$-Vector of a Gorenstein Toric Ring of a Compressed Polytope

A compressed polytope is an integral convex polytope all of whose pulling triangulations are unimodular. A $(q - 1)$-simplex $\Sigma$ each of whose vertices is a vertex of a convex polytope ${\cal P}$ is said to be a special simplex in ${\cal P}$ if each facet of ${\cal P}$ contains exactly $q - 1$ of the vertices of $\Sigma$. It will be proved that there is a special simplex in a compressed polytope ${\cal P}$ if (and only if) its toric ring $K[{\cal P}]$ is Gorenstein. In consequence it follows that the $h$-vector of a Gorenstein toric ring $K[{\cal P}]$ is unimodal if ${\cal P}$ is compressed.