Zaslavsky's theorem

Remarks

The theorem is due to Thomas Zaslavsky, in 1975. Proof idea: fix $H \in \mathcal{A}$, and let $\mathcal{A}'$ be $\mathcal{A}$ without $H$ and $\mathcal{A}''$ the arrangement induced on $H$. Each region of $\mathcal{A}'$ is either untouched by $H$ or cut in two, and the cut regions correspond to regions of $\mathcal{A}''$, so $r(\mathcal{A}) = r(\mathcal{A}') + r(\mathcal{A}'')$. The characteristic polynomial satisfies the matching recursion $\chi_{\mathcal{A}} = \chi_{\mathcal{A}'} - \chi_{\mathcal{A}''}$, and induction on the number of hyperplanes finishes the proof; the second count is similar.

Example: $n$ lines in general position in $\R^2$, no two parallel and no three through a point. Here $\chi(t) = t^2 - n t + \binom{n}{2}$, so $r = 1 + n + \binom{n}{2}$ and $b = 1 - n + \binom{n}{2} = \binom{n-1}{2}$. For $n = 3$ this gives $7$ regions, $1$ of them bounded.

Example: the braid arrangement of the hyperplanes $x_i = x_j$, $1 \le i < j \le n$, in $\R^n$ has $\chi(t) = t(t-1)\cdots(t-n+1)$, so it has $n!$ regions, one for each ordering of the coordinates.

Going deeper

For a student

The theorem turns a geometric counting problem into a computation in the intersection poset, through its Möbius function. For a central arrangement, $L(\mathcal{A})$ is a geometric lattice, and $\chi_{\mathcal{A}}$ is the characteristic polynomial of the associated matroid. The graphic arrangement of a graph $G$ on $n$ vertices has the hyperplanes $x_i = x_j$ for the edges $ij$. Its characteristic polynomial is the chromatic polynomial $\chi_G$ of $G$. So the theorem says that $(-1)^n \chi_G(-1)$ counts the acyclic orientations of $G$, a result of Stanley. When the hyperplanes have integer equations, $\chi_{\mathcal{A}}(q)$ counts the points of $\F_q^d$ off the reduced hyperplanes for every large enough prime $q$, which gives another route to computing it.