Symmetric tilings in the square lattice

M. Muzika Dizdarevi\' c

Abstract: We apply the method of Gr\"{o}bner bases to polyomino tilings, following and developing the ideas of Bodini and Nouvel. The emphasis is, in the spirit of the paper M. Muzika Dizdarevi\'{c}, R.~T. \v{Z}ivaljevi\'{c}, \emph{Symmetric polyomino tilings, tribones, ideals and Gr\"{o}bner bases}, Publ. Inst. Math. \textbf{98} (112) (2015), 1--23., on tiling problems with added symmetry conditions. The main problem studied in the paper covers case of tiling by three-in-line polyominoes, centrally symmetric with respect to the origin.

Keywords: Symmetric Z-tiling; ring of invariants; Gr\"{o}bner bases.

Classification (MSC2000): 52C20; 13P10

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