Signed polyomino tilings by n-in-line polyominoes and Gröbner bases

Abstract

Conway and Lagarias observed that a triangular region T(m) in a hexagonal lattice admits a signed tiling by three-in-line polyominoes (tribones) if and only if m 2 {9d?1, 9d}d2N. We apply the theory of Gr?bner bases over integers to show that T(m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m 2 {dn2 ? 1, dn2}d2N. Explicit description of the Gr?bner basis allows us to calculate the ?Gr?bner discrete volume? of a lattice region by applying the division algorithm to its ?Newton polynomial?. Among immediate consequences is a description of the tile homology group for the n-in-line polyomino. [Projekat Ministarstva nauke Republike Srbije, br. 174020 i br. 174034]

Signed Tilings by Ribbon L n-Ominoes, n Even, via Gröbner Bases

“…So in order to check that we have a Gröbner basis, we do not need to use G-polynomials. Due to the symmetry, some formulas above follow immediately from others: (14) follows from (12), (15) follows from (11), (16) follows from (13), and second formula in (18) follows from the first. For the rest, note that the leading monomial in Figure 12.…”

Gröbner Lattice-Point Enumerators and Signed Tiling by k-in-line Polyominoes

“…The approach to signed polyomino tilings via Gröbner bases was originally proposed by Bodini and Nouvel [5]. We independently discovered this idea and, inspired by [12], applied it in [10] to the calculation of tile homology groups (originally introduced in [12]) and in [9] for the study of Z-tilings with symmetries [9].…”

“…Since we apply the general theory to polynomials with integer coefficients, we work with strong Gröbner bases [1,11] (called a D-Gröbner base in [4]), see also [10,Section 5] or our Section 6 for a brief introduction. 1.3.…”

Homology of polyomino tilings on flat surfaces

“…We define a homomorphism φ : ⟨Θ⟩ → Q/Z with φ(Θ) ̸ = 0. If Θ has infinite order we set φ(Θ) = 1 2 mod Z, while if Θ has finite order n > 1, then we define φ(Θ) = Reid's tiling homology group was systematically studied using Gröbner bases in the works of Muzika-Dizdarević, Timotijević andŽivaljević, see [15] and [16].…”