A Polyomino Tiling Problem Explored via Gröbner Bases

A formal research investigation into the algebraic characterisation of tiling rectangular regions using thick L-shaped polyominoes. The project models polyomino tiles and regions as bivariate polynomials and applies Gröbner basis theory to solve the ideal membership problem: whether a given rectangle can be tiled by a fixed tile. The work establishes necessary and sufficient conditions for tileability under various modular constraints on the tile size n, and proves three main theorems for the even-even, mixed-parity, and congruence-based cases. It also includes original Python implementations for symbolic verification, and manual derivations of S-polynomials to confirm the basis structure. This project combines combinatorics, commutative algebra, and symbolic computation, and demonstrates the use of Gröbner bases in solving geometric problems.