Authors: Dizdarević, Manuela Muzika
Timotijević, Marinko
Živaljević, Rade 
Title: Signed polyomino tilings by n-in-line polyominoes and Gröbner bases
Journal: Publications de l'Institut Mathematique
Volume: 99
Issue: 113
First page: 31
Last page: 42
Issue Date: 1-Jan-2016
Rank: M24
ISSN: 0350-1302
DOI: 10.2298/PIM1613031M
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 ∈ 2 {9d-1, 9d}d∈N. We apply the theory of Gröbner bases over integers to show that T(m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m ∈ {dn2 - 1, dn2}d∈N. Explicit description of the Gröbner basis allows us to calculate the 'Gröbner 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.
Keywords: Gröbner bases | Signed polyomino tilings
Publisher: Mathematical Institute of the SASA
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Topology, geometry and global analysis on manifolds and discrete structures 

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