20.500.12556/RUL-161810 Tlakovanje trikotnika s polinominami delo diplomskega seminarja Signed polyomino tilings of a triangular area Problem tlakovanja je geometrijski problem pri katerem želimo določen lik pokriti z vnaprej podanimi ploščicami. Ker so tovrstni problemi v splošnem težki, smo definirali označena pokritja, ki nam podajajo potreben pogoj za obstoj tlakovanja. V diplomskem delu smo opisali, kako preidemo iz geometrijskega na algebraičen problem. Našli smo izomorfizem, ki slika ploščice v polinome in dokazali, da je problem iskanja označenega pokritja ekvivalententen problemu vsebovanosti polinoma v idealu. Dokazali smo, da je polinom vsebovan v idealu natanko takrat, ko se reducira v 0 po modulu Gröbnerjeve baze. Za iskanje Gröbnerjeve baze ideala, smo uporabili Buchbergerjev algoritem. Na koncu smo na trikotnem mrežastem območju uporabili izpeljano teorijo in dokazali izrek Conwaya in Lagariasa. Tiling problem is a geometric problem in which we aim to cover a certain figure with pre-defined tiles. Since such problems are generally challenging, we have defined signed tilings, which give us a necessary condition for the existence of a tiling. In the thesis, we described how we transition from a geometric problem to an algebraic one. We found an isomorphism that maps tiles to polynomials and proved that the problem of finding a signed tiling is equivalent to the problem of polynomial containment in an ideal. We proved that a polynomial is contained in an ideal exactly when it reduces to 0 with respect to the Gröbner basis. To find the Gröbner basis of the ideal, we used Buchberger's algorithm. Finally, on a triangular lattice region, we applied the derived theory and proved Conway's and Lagarias's theorem. celica polinomina polinom ploščica tlakovanje označeno pokritje mrežasto območje kolobar redukcija ideal sizigija nasičenost Gröbnerjeva baza Buchbergerjev algoritem cell polyomino polynomial tile tiling signed tiling lattice ring reduction ideal syzygy saturation Gröbner basis Buchberger algorithm true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2024-09-14 08:15:05 2024-09-14 08:15:08 2024-09-17 16:53:42 0000-00-00 00:00:00 2024 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.1 143349 207925251 14075.pdf 14075.pdf 1 1A07FA5DF7294AB96364EB18D46CF3FD db8ec7335b4e4f529609f47a2f801d6c227eff353b0b5f7ddabeac2b1f5e9700 9da4d21b-7260-11ef-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=190794 Fakulteta za matematiko in fiziko 0 0 0