A study of the algebraic varieties with an algebraic group action

2014 Fiscal Year Final Research Report

• Project/Area Number: 23540057
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Denki University
• Principal Investigator: NAKANO Tetsuo 東京電機大学, 理工学部, 教授 (00217796)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: Booleanグレブナ基底 / 井上アルゴリズム / 数独型パズル

Outline of Final Research Achievements

In this research, we have studied the Inoue algorithm, which is a very efficient method for solving a system of the Boolean polynomial equations, and its application to the puzzles of Sudoku type. We have firstly defined the Inoue invariant of such a system, which is the triple data of the minimum tree diagram among all the trees appearing in the process of the generalized Inoue algorithm. As an application, using the formulation of the puzzles of Sudoku type in terms of the Boolean polynomial equations, we have classified the Inoue invariant of the 4-doku and the diagonal 5-doku puzzles. It turns out that in these cases, every puzzle with a unique solution has a trivial Inoue invariant (2,1,1) except 2 special types of diagonal 5-doku puzzles. We have also shown by experiments that, in the case of Sudoku puzzles (9 times 9), the Inoue invariant is an excelent indicator of the difficulty of the puzzles.

Free Research Field

代数幾何学，計算代数学