Gröbner Basis Representations of Sudoku

Linear algebraic approach to Gröbner–Shirshov basis theory

We construct a new efficient algorithm for finding Gröbner–Shirshov bases for noncommutative algebras and their representations. This algorithm uses the Macaulay matrix [F.S. Macaulay, On some formula in elimination, Proc. London Math. Soc. 33 (1) (1902) 3–27], and can be viewed as a representation theoretic analogue of the F4 algorithm developed by J.C. Faugère. We work out some examples of un...

Moduli of Mckay Quiver Representations Ii: Gröbner Basis Techniques

In this paper we introduce several computational techniques for the study of moduli spaces of McKay quiver representations, making use of Gröbner bases and toric geometry. For a finite abelian group G ⊂ GL(n, k), let Yθ be the coherent component of the moduli space of θ-stable representations of the McKay quiver. Our two main results are as follows: we provide a simple description of the quiver...

Sudoku Associated Two Dimensional Bijections for Image Scrambling

Sudoku puzzles are now popular among people in many countries across the world with simple constraints that no repeated digits in each row, each column, or each block. In this paper, we demonstrate that the Sudoku configuration provides us a new alternative way of matrix element representation by using block-grid pair besides the conventional row-column pair. Moreover, we discover six more matr...

Automatic Geometric Theorem Proving using Gröbner Bases

Meta-model-based implementation of Sudoku: Eclipse vs. Visual Studio

By considering Sudoku as a language, where a Sudoku puzzle is an instance of the language, we are able to apply meta-model-based technologies for the implementation of Sudoku, including correctness checking of a puzzle and solving strategies. The description of Sudoku includes not only the structure of Sudoku, but also covers constraints, textual representation, graphical representation, and be...