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

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.