[Paper Review] The no-three-in-line problem on a torus

Jim Fowler, Andrew Groot+2 more|arXiv (Cornell University)|Mar 29, 2012

Geometric and Algebraic Topology6 references4 citations

TL;DR

This paper investigates the no-three-in-line problem on discrete tori, redefining lines as cosets of cyclic subgroups in $ℤ_m \times \u2124_n$. Using group theory and Gröbner bases, it establishes exact values for maximal point placements: $T(\mathbb{Z}_p \times \mathbb{Z}_{p^2}) = 2p$ and $T(\mathbb{Z}_p \times \mathbb{Z}_{pq}) = p+1$ for distinct primes $p$, $q$, and computes $T(\mathbb{Z}_m \times \mathbb{Z}_n)$ for $2 \leq m \leq 7$, $2 \leq n \leq 19$ via Hilbert series in Macaulay2.

ABSTRACT

Let $T(\Z_m imes \Z_n)$ denote the maximal number of points that can be placed on an $m imes n$ discrete torus with "no three in a line," meaning no three in a coset of a cyclic subgroup of $\Z_m imes \Z_n$. By proving upper bounds and providing explicit constructions, for distinct primes $p$ and $q$, we show that $T(\Z_p imes \Z_{p^2}) = 2p$ and $T(\Z_p imes \Z_{pq}) = p+1$. Via Gröbner bases, we compute $T(\Z_m imes \Z_n)$ for $2 \leq m \leq 7$ and $2 \leq n \leq 19$.

Motivation & Objective

• To determine the maximum number of points that can be placed on an $m \times n$ discrete torus with no three in a line, where lines are defined as cosets of cyclic subgroups of $\mathbb{Z}_m \times \mathbb{Z}_n$.
• To establish exact values of $T(\mathbb{Z}_m \times \mathbb{Z}_n)$ for specific torus dimensions, particularly for prime and composite orders.
• To develop and apply computational algebraic methods, including Gröbner bases and Hilbert series, to enumerate solutions and verify bounds.
• To compare solutions on tori with those on planar lattices, revealing that toroidal constraints reduce maximal solution sizes.

Proposed method

• Define lines on the torus as images of lines in $\mathbb{Z} \times \mathbb{Z}$ under the covering map, corresponding to cosets of maximal cyclic subgroups of $\mathbb{Z}_m \times \mathbb{Z}_n$.
• Use group-theoretic analysis to bound $T(\mathbb{Z}_m \times \mathbb{Z}_n)$, the maximum number of points with no three in a line, by studying cyclic subgroup structure.
• Apply Gröbner bases over finite fields to model the no-three-in-line condition as an ideal in a polynomial ring, encoding point placements and forbidden configurations.
• Compute the Hilbert series of the quotient ring $R = K[x_{i,j}]/I$, where $I$ encodes the no-three-in-line constraints, to enumerate solutions of each size.
• Use the computer algebra system Macaulay2 to compute Hilbert functions and series, extracting the degree of the highest non-zero term as the maximal solution size.
• Verify results empirically by constructing explicit point sets achieving the computed bounds, especially for $\mathbb{Z}_p \times \mathbb{Z}_{p^2}$ and $\mathbb{Z}_p \times \mathbb{Z}_{pq}$.

Experimental results

Research questions

• RQ1 What is the maximum number of points that can be placed on a discrete torus $\mathbb{Z}_m \times \mathbb{Z}_n$ such that no three lie on a coset of a cyclic subgroup?
• RQ2 How do the maximal solutions on tori compare in size and number to those on planar $n \times n$ lattices?
• RQ3 Can exact formulas be derived for $T(\mathbb{Z}_p \times \mathbb{Z}_{p^2})$ and $T(\mathbb{Z}_p \times \mathbb{Z}_{pq})$ for distinct primes $p$, $q$?
• RQ4 What computational methods can be used to systematically compute $T(\mathbb{Z}_m \times \mathbb{Z}_n)$ for small $m$, $n$?
• RQ5 Why does the $14 \times 14$ torus allow only 12 points, despite its size, and what does this reveal about toroidal constraints?

Key findings

• For distinct primes $p$ and $q$, the maximal number of points on a $\mathbb{Z}_p \times \mathbb{Z}_{p^2}$ torus with no three in a line is exactly $2p$.
• For distinct primes $p$ and $q$, the maximal number of points on a $\mathbb{Z}_p \times \mathbb{Z}_{pq}$ torus is $p+1$.
• The value $T(\mathbb{Z}_2 \times \mathbb{Z}_{2n}) = 4$ holds for all positive integers $n$, established via explicit construction.
• The $14 \times 14$ discrete torus allows only 12 points with no three in a line, an anomaly where solution size is less than the torus size.
• For $2 \leq m \leq 7$ and $2 \leq n \leq 19$, the paper computes $T(\mathbb{Z}_m \times \mathbb{Z}_n)$ explicitly using Gröbner bases and Hilbert series.
• The number of maximal solutions on tori is generally much smaller than on planar lattices, with odd tori showing closer correspondence to lattice solutions than even tori.