The first author is grateful to Antonio Caruso for his useful discussions on and help for computer science problems during the beautiful period spent at the Department of Mathematics in Lecce. He would also like to thank the Department of Mathematics and Applications in Milano for the nice welcome.

Preliminaries {#sec:preliminaries}
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Hyperplane arrangements and Salvetti’s complex
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In this section we will recall some basic notions from the theory of hyperplanes arrangements. The interested reader is referred to, for instance, Orlik and Terao [@OT92] for a much more detailed and extended study.

#### Hyperplane arrangements

In geometry and combinatorics, an [*arrangement of hyperplanes*]{} is a finite set ${\mathcal{A}}$ of hyperplanes in a linear, affine, or projective space $S$. The [*cardinality*]{} $\left|{\mathcal{A}}\right|$ of the arrangement ${\mathcal{A}}$ is the number of hyperplanes in ${\mathcal{A}}$.

One is normally interested both in the real and in the complex case, hence let ${\mathbb{K}}$ be either ${\mathbb{R}}$ or ${\mathbb{C}}$ and let $V$ be either ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$. Thus, given the canonical base $\{e_1,\dotsc ,e_n\}$ in $V$, each hyperplane $H \in {\mathcal{A}}$ is the kernel of a degree-1 polynomial $\alpha_H \in {\mathbb{K}}[x_1,\dotsc, x_n]$, defined up to a constant. The product $${\mathcal{Q}}({\mathcal{A}})= \prod_{H \in {\mathcal{A}}} \alpha_H$$ is 