Dehn–Sommerville equations

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.

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 ${\displaystyle \left[{\frac {d+1}{2}}\right]}$ 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.

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