The cyclicity rank of empty lattice simplices

TL;DR

This paper investigates the cyclicity rank of empty lattice simplices, establishing the maximum possible rank for dimensions up to 8 and analyzing its asymptotic behavior.

Contribution

It determines the maximal cyclicity rank of empty lattice simplices for dimensions up to 8 and explores its asymptotic growth.

Findings

Maximal cyclicity rank for dimensions up to 8 is identified.

Asymptotic behavior of cyclicity rank is characterized.

Empty lattice simplices are cyclic for dimensions up to 4.

Abstract

We are interested in algebraic properties of empty lattice simplices $\Delta$, that is, $d$-dimensional lattice polytopes containing exactly $d+1$ points of the integer lattice $\mathbb{Z}^d$. The cyclicity rank of $\Delta$ is the minimal number of cyclic subgroups that the quotient group of $\Delta$ splits into. It is known that up to dimension $d \leq 4$, every empty lattice $d$-simplex is cyclic, meaning that its cyclicity rank is at most $1$. We determine the maximal possible cyclicity rank of an empty lattice $d$-simplex for dimensions $d \leq 8$, and determine the asymptotics of this number up to a logarithmic term.