Hermite normal forms and $δ$-vector

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

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