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 fi denote the number of i-dimensional faces of P. The sequence
f(P)=(f0,f1,\ldots,fd-1)
is called the f-vector of the polytope P. Additionally, set
f-1=1,fd=1.
Then for any k = -1, 0, ..., d - 2, the following Dehn–Sommerville equation holds:
| d-1 | |
| \sum | |
| j=k |
(-1)j\binom{j+1}{k+1}fj=(-1)d-1fk.
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.
See main article: h-vector.
Sommerville found a different way to state these equations:
| k-1 | |
| \sum | |
| i=-1 |
(-1)d+i\binom{d-i-1}{d-k}fi=
| d-k-1 | |
| \sum | |
| i=-1 |
(-1)i\binom{d-i-1}{k}fi,
where 0 ≤ k ≤ (d-1). This can be further facilitated introducing the notion of h-vector of P. For k = 0, 1, ..., d, let
hk=
| k | |
| \sum | |
| i=0 |
(-1)k-i\binom{d-i}{k-i}fi-1.
The sequence
h(P)=(h0,h1,\ldots,hd)
is called the h-vector of P. The f-vector and the h-vector uniquely determine each other through the relation
| d | |
| \sum | |
| i=0 |
fi-1(t-1)d-i
| d | |
| =\sum | |
| k=0 |
hktd-k.
Then the Dehn–Sommerville equations can be restated simply as
hk=hd-k for0\leqk\leqd.
The equations with 0 ≤ k ≤ (d-1) are independent, and the others are manifestly equivalent to them.
Richard Stanley gave an interpretation of the components of the h-vector of a simplicial convex polytope P in terms of the projective toric variety X associated with (the dual of) P. Namely, they are the dimensions of the even intersection cohomology groups of X:
hk=\dimQ\operatorname{IH}2k(X,Q)
(the odd intersection cohomology groups of X are all zero). In this language, the last form of the Dehn–Sommerville equations, the symmetry of the h-vector, is a manifestation of the Poincaré duality in the intersection cohomology of X.