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.
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