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The cyclicity rank of empty lattice simplices

Lukas Abend, Matthias Schymura

TL;DR

This work investigates the cyclic structure of quotient groups associated with empty lattice simplices by introducing the cyclicity rank $cr(\Delta)$ and its dimension-maximizing variant $cr_e(d)$. The authors develop reductions to $p$-power simplices via Hermite normal form, establishing sharp characterizations of $cr(\Delta)$ in that setting and deriving a sequence of bounds and exact values. They provide exact results for small dimensions (notably $d\le 8$) and derive asymptotic behavior $cr_e(d)=d-\Theta(\log d)$ with explicit upper bounds, including the general bound $cr_p(d) \le d-\lfloor \log_p d\rfloor-1$ for primes $p$. The paper also demonstrates that for $p=3$ one can exceed the 2-power case in large dimensions and offers constructions and computational verifications (via Sage) for specific high-dimensional empty $3$-power simplices, while highlighting open questions about monotonicity in $p$ and the boundedness of the prime achieving $cr_e(d)$.

Abstract

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 $\mathbb{Z}^d$. 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 $d \leq 4$, 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.

The cyclicity rank of empty lattice simplices

TL;DR

This work investigates the cyclic structure of quotient groups associated with empty lattice simplices by introducing the cyclicity rank and its dimension-maximizing variant . The authors develop reductions to -power simplices via Hermite normal form, establishing sharp characterizations of in that setting and deriving a sequence of bounds and exact values. They provide exact results for small dimensions (notably ) and derive asymptotic behavior with explicit upper bounds, including the general bound for primes . The paper also demonstrates that for one can exceed the 2-power case in large dimensions and offers constructions and computational verifications (via Sage) for specific high-dimensional empty -power simplices, while highlighting open questions about monotonicity in and the boundedness of the prime achieving .

Abstract

We are interested in algebraic properties of empty lattice simplices , that is, -dimensional lattice polytopes containing exactly points of the integer lattice . The cyclicity rank of is the minimal number of cyclic subgroups that the quotient group of splits into. It is known that up to dimension , every empty lattice -simplex is cyclic, meaning that its cyclicity rank is at most . We determine the maximal possible cyclicity rank of an empty lattice -simplex for dimensions , and determine the asymptotics of this number up to a logarithmic term.
Paper Structure (8 sections, 23 theorems, 53 equations, 1 figure, 1 table)

This paper contains 8 sections, 23 theorems, 53 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Quotient Groups of Lattice Simplices and Problem Statement
  3. Reductions and p-power simplices
  4. The Hermite normal form of p-power simplices
  5. Small Dimensions and the Case p=2
  6. The maximal cyclicity rank of empty 2-power simplices
  7. Large dimensions and the case p=3
  8. Some results for empty 3-power simplices

Key Result

Proposition 1

Let $\Delta,\Delta' \subseteq \mathbb{R}^d$ be lattice simplices which both have the origin as a vertex. Then, we have for some permutation matrix $P \in \mathbb{Z}^{d \times d}$ and some $k \in \{0,1,\ldots,d\}$, where $U_k \in \mathbb{Z}^{d \times d}$ is the unimodular matrix whose $(i,k)$-entries equal $-1$, for all $1 \leq i \leq d$, whose $(i,i)$-entries equal $1$, for all $1 \leq i \leq d$

Figures (1)

  • Figure 1: The triangle $\Delta = \mathop{\mathrm{conv}}\nolimits\{\mathbf{0},\binom{4}{0},\binom{0}{3}\}$ on the left is cyclic with $G_\Delta \cong \mathbb{Z}_{12}$ generated by $\binom{1}{1}$; the red lattice point. The triangle $\Delta' = \mathop{\mathrm{conv}}\nolimits\{\mathbf{0},\binom{3}{0},\binom{0}{3}\}$ however is not cyclic, and has $G_{\Delta'} \cong \mathbb{Z}_3 \times \mathbb{Z}_3$ with generators $\binom{1}{0}$ and $\binom{0}{1}$; the two blue lattice points.

Theorems & Definitions (47)

  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 3: Cyclicity rank of a simplex
  • Proposition 4
  • ...and 37 more