I know that the number of bounded regions of a homogeneous hyperplane arrangement $\mathcal{A}$(a collection of n hyperplanes) in $\mathbb R^d$is ${ n - 1 \choose d}$ but how can this be expressed in terms of the characteristic polynomial of the hyperplane arrangement? I saw that this might be equal to $|p(1)|$ where $p$ is the characteristic polynomial of $\mathcal{A}.$ Is there any sort of explanation why this is correct?
Any clarification will be greatly appreciated!
