Hermite normal forms and $\delta$-vector

TL;DR

This paper classifies possible $oldsymbol{ ext{delta}}$-vectors of integral polytopes with sum up to 4, using Hermite normal forms to analyze integral simplices and extend previous classifications.

Contribution

It provides a complete classification of $oldsymbol{ ext{delta}}$-vectors with sum 4, expanding prior work limited to sum up to 3, through Hermite normal form analysis of simplices.

Findings

Classified all $oldsymbol{ ext{delta}}$-vectors with sum 4.

Established that all such $oldsymbol{ ext{delta}}$-vectors can be realized by simplices.

Extended the characterization of $oldsymbol{ ext{delta}}$-vectors to sum 4.

Abstract

Let $\delta(\Pc) = (\delta_0, \delta_1,..., \delta_d)$ 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 $\sum_{i=0}^d \delta_i \leq 3$, 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 $\sum_{i=0}^d \delta_i \leq 4$, by means of Hermite normal forms of square matrices.