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I have a family of hyperplane arrangements, and I'd like to describe their characteristic polynomials. When the hyperplanes are defined over the integers, the easiest way for me to do this is to use the theorem of Athanasiadis (in "Characteristic Polynomials of Subspace Arrangements and Finite Fields") which states that, for sufficiently large primes $q$, the characteristic polynomial $\chi(q)$ counts the number of points in the complement of the arrangement over $\mathbb{F}_q$.

However, in full generality, my hyperplanes may be defined by equations with arbitrary real coefficients. I'd like to know if, in this context, there is still a way to describe the characteristic polynomial at infinitely many values in terms of point counting.

My advisor has suggested that I can adjoin the coefficients of the hyperplanes to $\mathbb{Z}$ to get a larger ring $R$, and then mimic Athanasiadis's proof using the finite fields which are quotients of $R$. However, as this is going into an entirely combinatorial paper, I'd like to avoid digressing into finicky algebraic arguments like this if possible. So is there a source I can cite for obtaining the characteristic polynomial of a general real arrangement in terms of point counting, similarly to the result of Athanasiadis? If not, what is an accessible source for the necessary ingredients of a proof?

EDIT: To be more specific, two facts which would be useful to have citations for in generalizing Athanasiadis's proof are 1) any finite type $\mathbb{Z}$-algebra of characteristic 0 has quotients which are fields of arbitrarily large characteristic and 2) those fields are finite. While I can include proofs of these things, I'd prefer to be able to outsource that work to other sources.

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  • $\begingroup$ Perhaps Theorem 9.1 of arxiv.org/abs/math/0309330, which is a general formula using the Möbius function for any valuation, is useful for you. $\endgroup$
    – Sam Hopkins
    Commented Sep 27, 2020 at 17:26
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    $\begingroup$ I would not call counting points over $\mathbb{F}_q$ "entirely combinatorial." This may be the case for hyperplanes like $x_i=\pm x_j$, where the arithmetic of finite fields is irrelevant, but once you have hyperplanes like $x_i+x_j+x_k=0$, then the counting will involve algebra. That is why, for instance, the number of regions of the resonance arrangement remains intractable. For nonintegral arrangements, the best tool seems to be Whitney's theorem. See for instance Sections 5 and 6 of www-math.mit.edu/~rstan/papers/deform.pdf. $\endgroup$
    – Richard Stanley
    Commented Sep 27, 2020 at 17:27

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