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]
We apply the theory of Gröbner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions T N = T 3k−1 and T N = T 3k in a hexagonal lattice admit a signed tiling by three-in-line polyominoes (tribones) symmetric with respect to the 120 • rotation of the triangle if and only if either N = 27r − 1 or N = 27r for some integer r 0. The method applied is quite general and can be adapted to a large class of symmetric tiling problems.
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