A challenge of solving the normality conjecture for cut polytopes affirmatively which yields a theoretical proof of the four color theorem

2015 Fiscal Year Final Research Report

• Project/Area Number: 25610032
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Osaka University
• Principal Investigator: HIBI Takayuki 大阪大学, 情報科学研究科, 教授 (80181113)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 切断多面体 / 切断イデアル / 正規多面体 / 正則単模三角形分割 / トーリックイデアル / グレブナー基底 / イニシャルイデアル / 実験計画

Outline of Final Research Achievements

The purpose of the present research was, following the established techniques due to the principal investigator together with his colleagues, to face a challenge of solving the normality conjecture for cut polytopes affirmatively. The reason why the normality conjecture for cut polytopes fascinates the combinatorialists is the fact, due to David E. Speyer, that the affirmative answer of the normality conjecture for cut polytopes guarantees the four color theorem. We have faced a challenge of solving the normality conjecture for cut polytopes affirmatively from the viewpoint of combinatorics on dilated cut polytopes as well as that of initial ideals of Gr\"obner bases of cut ideals. Furthermore, with taking into account the negative answer of the normality conjecture for cut polytopes, we have developed the various studies on cut polytopes and cut ideals.

Free Research Field

計算可換代数と組合せ論