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In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their general form was established by Duncan Sommerville in 1927. The Dehn–Sommerville equations can be restated as a symmetry condition for the ''h''-vector'' of the simplicial polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for simple polytopes. == Statement == Let ''P'' be a ''d''-dimensional simplicial polytope. For ''i'' = 0, 1, ..., ''d''−1, let ''f''''i'' denote the number of ''i''-dimensional faces of ''P''. The sequence : is called the ''f''-vector of the polytope ''P''. Additionally, set : Then for any ''k'' = −1, 0, …, ''d''−2, the following Dehn–Sommerville equation holds: : When ''k'' = −1, it expresses the fact that Euler characteristic of a (''d'' − 1)-dimensional simplicial sphere is equal to 1 + (−1)''d''−1. Dehn–Sommerville equations with different ''k'' are not independent. There are several ways to choose a maximal independent subset consisting of equations. If ''d'' is even then the equations with ''k'' = 0, 2, 4, …, ''d''−2 are independent. Another independent set consists of the equations with ''k'' = −1, 1, 3, …, ''d''−3. If ''d'' is odd then the equations with ''k'' = −1, 1, 3, …, ''d''−2 form one independent set and the equations with ''k'' = −1, 0, 2, 4, …, ''d''−3 form another. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dehn–Sommerville equations」の詳細全文を読む スポンサード リンク
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