Algebraic Properties and Invariants of Polyominoes

Polyominoes are two-dimensional objects obtained by joining edge by edge squares of same size. Originally, polyominoes appeared in mathematical recreations, but it turned out that they have applications in various fields, for example, theoretical physics and bio-informatics. Among the most popular topics in combinatorics related to polyominoes one finds enumerating polyominoes of given size, including the asymptotic growth of the numbers of polyominoes, tiling problems, and reconstruction of polyominoes. Recently Qureshi introduced a binomial ideal induced by the geometry of a given polyomino, called polyomino ideal, and its related algebra. From that moment different authors studied algebraic properties and invariants related to this ideal, such as primality, Gröbner bases, Gorensteinnes and Castelnuovo-Mumford regularity. In this thesis, we provide an overview on the results that we obtained about polyomino ideals and its related algebra. In the first part of the thesis, we discuss questions about the primality and the Gröbner bases of the polyomino ideal. In the second part of the thesis, we talk over the Castelnuovo-Mumford regularity, Hilbert series, and Gorensteinnes of the polyomino ideal and its coordinate ring.