Hermite normal forms and $\delta$-vector

Abstract

Let $δ(\Pc) = (δ_0, δ_1,..., δ_d)$ be the $\delta$-vector of an integral polytope $\Pc \subset \RR^N$ of dimension $d$. Following the previous work of characterizing the $\delta$-vectors with ${\sum }_{i=0}^d{\delta }_i\le 3$, the possible $\delta$-vectors with ${\sum }_{i=0}^d{\delta }_i=4$ will be classified. And each possible $\delta$-vectors can be obtained by simplices. We get this result by studying the problem of classifying the possible integral simplices with a given $\delta$-vector $({\delta }_0,{\delta }_1,...,{\delta }_d)$, where ${\sum }_{i=0}^d{\delta }_i\le 4$, by means of Hermite normal forms of square matrices.