Combinatorial Theory

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.

Mathematics Subject Classifications: 52B20, 11H06, 14E30

Keywords: Empty lattice simplices, quotient groups, Hermite normal forms