A Polyomino Tiling Problem Explored via Gröbner Bases
Year: 2025
Type: article
Abstract: 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.
Author: Salman, Mustafa
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6931004147
FWCI:
Cites:
Cited by:
Related to: