Sudokus and Gröbner Bases: Not Only a Divertimento
- 463 Downloads
Sudoku is a logic-based placement puzzle. We recall how to translate this puzzle into a 9-colouring problem which is equivalent to a (big) algebraic system of polynomial equations. We study how far Gröbner bases techniques can be used to treat these systems produced by Sudokus. This general purpose tool can not be considered as a good solver, but we show that it can be useful to provide information on systems that are —in spite of their origin— hard to solve.
KeywordsPolynomial Equation Algebraic System Polynomial System Hilbert Scheme Colouring Problem
Unable to display preview. Download preview PDF.
- 2.Bayer, D.: The division algorithm and the Hilbert scheme. Ph. D. Thesis. Harvard University (June 1982)Google Scholar
- 3.Becker, T., Weispfenning, V.: Gröbner bases. Graduate Texts in Mathematics, vol. 141. Springer, New York (1993); A computational approach to commutative algebra, In cooperation with Heinz KredelGoogle Scholar
- 4.Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms. Springer, Berlin (1997)Google Scholar
- 6.Felgenhauer, B., Jarvis, F.: Enumerating possible sudoku grids (2005) (online, accessed December 30, 2005), http://www.afjarvis.staff.shef.ac.uk/sudoku/sudoku.pdf
- 7.Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern (2005), http://www.singular.uni-kl.de
- 9.Kelley, C.T.: Iterative methods for linear and nonlinear equations. Frontiers in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1995); With separately available softwareGoogle Scholar
- 10.Knuth, D.E.: Dancing links (preprint)Google Scholar
- 12.Pegg, E.: Sudoku variations (2005) (online, accessed December 12, 2005), http://www.maa.org/editorial/mathgames/mathgames_09_05_05.html
- 13.Sturmfels, B.: Solving Systems of Polynomial Equations. Amer. Math. Soc., CBMS Regional Conferences Series, No. 97, Providence, Rhode Island (2002)Google Scholar