# PRIMENA GREBNEROVIH BAZA NA PROBLEME POPLOÄŒAVANJA

Title: | PRIMENA GREBNEROVIH BAZA NA PROBLEME POPLOÄŒAVANJA |

Author: | Muzika DizdareviÄ‡, Manuela |

Abstract: | Subject of this doctoral thesis is the application of algebraic techniques on one of the central topics of combinatorics and discrete geometry - polyomino tiling. Polyomino tilings are interesting not only to mathematicians, but also to physicists and biologists, and they can also be applied in computer science. In this thesis we put some emphasis on possibility to solve special class of tiling problems, that are invariant under the action of nite group, by using theory of Gr obner basis for polynomial rings with integer coe cients. Method used here is re ecting deep connection between algebra, geometry and combinatorics. Original scienti c contribution of this doctoral thesis is, at the rst place, in developing a techniques which enable us to consider not only ordinary Z?tiling problems in a lattice but the problems of tilings which are invariant under some subgroups of the symmetry group of the given lattice. Besides, it provides additional generalizations, originally provided by famous mathematicians J. Conway and J. Lagarias, about tiling of the triangular region in hexagonal lattice. Here is a summary of the content of the theses. In the rst chapter we give an exposition of the Gr obner basis theory. Especially, we emphasize Gr obner basis for polynomial rings with integer coe cients. This is because, in this thesis, we use algorithms for determining Gr obner basis for polynomials with integer coe cients. Second chapter provides basic facts about regular lattices in the plane. Also, this chapter provides some fundamental terms of polyomino tiling in the square and hexagonal lattice. Third chapter of this thesis is about studying Ztilings in the square lattice, which are invariant under the subgroup G of the group of all isometric transformations of the lattice which is generated by the central symmetry. One of the steps to resolve this problem was to determine a ring of invariants PG and its generators and relations among them. We use Gr obner basis theory to achieve this. Forth chapter covers the analysis of Ztilings in the hexagonal lattice which are symmetric with respect to the rotation of the plane for the angle of 120 . Main result of the fourth chapter is the theorem which gives conditions for symmetric tiling of the triangular region in plane TN, where N is the number of hexagons on each side of triangle. This theorem is one of the possible generalizations of the well known result, provided by Conway and Langarias. Fifth chapter provides another generalization of Conway and Lagarias result, but this time it is about determining conditions of tiling of triangular region TN in the hexagonal lattice not only with tribones, but with nbones. nbone is basic shape of of n connected cells in the hexagonal lattice, where n is arbitrary integer. |

URI: | http://hdl.handle.net/123456789/4503 |

Date: | 2017 |

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