2024/07/01 by Lukas Abend, Abend, Lukas, Matthias Schymura +1 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Rings, Modules, and Algebras #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2407.01179
We are interested in algebraic properties of empty lattice simplices Δ, that is, d-dimensional lattice polytopes containing exactly d+1 points of the integer lattice ℤd. The cyclicity rank of Δ is the minimal number of cyclic subgroups that the quotient group of Δ splits into. It is known that up to dimension d ≤ 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 ≤ 8, and determine the asymptotics of this number up to a logarithmic term.