# connected components of a real hyperplane arrangement

Let us consider the following configuration of hyperplanes in the real vector space V with coordinates $z_1,\ldots,z_n$: the hyperplanes are numbered by all the nonempty subsets $J\subset I=\{1,\ldots,n\}$, and the hyperplane $H_J$ is given by $\sum_{i\in J}z_i=0$.

Question: How many connected components does the complement $V\setminus\cup_{J}H_J$ have? That is, what are they naturally numbered by?

• This is a trivial observation, but if you use the finite field method, you get the characteristic polynomial is $\chi(q) = q^n - |\{(a_1,\ldots,a_n) \in \mathbb{F}_q^n\colon \textrm{some nonempty subset of the$a_i$sums to$0 \mod q$}\}|$. That looks like a terribly difficult counting problem, of course. Commented Jan 15, 2013 at 17:06

This is not known as far as I know, and seems to be a hard problem, see the references and comments in this MO question.

There is a formula for the number of components which unfortunately is pretty useless, namely, $$\sum_{k,r} \frac{(-1)^{n+k+r}}{k!}f(k,n,r),$$ where $f(k,n,r)$ is the number of real $k\times n$ $(0,1)$-matrices of rank $r$ with no zero row and no two rows equal.