# 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.

## 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.

## Equivalent formulations

Sommerville found a different way to state these equations:

where 0 ≤ k ≤ ½(d−1). This can be further facilitated introducing the notion of *h*-vector of *P*. For *k* = 0, 1, …, *d*, let

The sequence

is called the *h*-vector of *P*. The *f*-vector and the *h*-vector uniquely determine each other through the relation

Then the Dehn–Sommerville equations can be restated simply as

The equations with 0 ≤ k ≤ ½(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*:

(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*.

## References

- Branko Grünbaum,
*Convex polytopes*. Second edition. Graduate Texts in Mathematics, 221, Springer, 2003 ISBN 0-387-00424-6 - Richard Stanley,
*Combinatorics and commutative algebra*. Second edition. Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1996. x+164 pp. ISBN 0-8176-3836-9 - Duncan Sommerville (1927) The relations connecting the angle sums and volume of a polytope in space of n dimensions Proceedings of the Royal Society Series A 115:103–19, weblink from JSTOR.
- G. Ziegler,
*Lectures on Polytopes*, Springer, 1998. ISBN 0-387-94365-X