Let $H_1,\ldots,H_n$ be hyperplanes in $\Bbb R^d$. Denote $\mathcal{H} :=\{H_1,\ldots,H_n\}$ and let $c(\mathcal{H})$ be the number of regions in the complement: $\Bbb R^d\setminus \cup H_i$.  

**Question:**  What is the complexity of computing $c(\mathcal{H})$?

Here we are assuming that $H_i$ are defined explicitly over $\Bbb Q$, and that the dimension $d$ is NOT bounded.  For a fixed $d$ there is plenty of literature, see e.g. Halperin-Sharir [Arrangements][1] survey.  As far as I can tell, none of that literature is applicable.  

Note that for *graphical arrangements* $\{x_i-x_j=0 : (ij)\in E\}$ corresponding to the graph $G=([n],E)$, the number of regions $c(\mathcal H)$ is an evaluation of the chromatic (and therefore Tutte) polynomial, and thus #P-hard.  See e.g. Welsh's ``[Complexity: knots, colouring and counting][2]''  book, Chapter 6. 

**Comment:** It feels like this should be well known, so maybe this is a reference request.  The problem is in PSPACE and feels similar to $\exists \Bbb R$ (see [Wikipedia page][3]), except it's a counting problem. Is it $\exists \Bbb R$-hard, for example?  


  [1]: http://www.csun.edu/~ctoth/Handbook/chap28.pdf
  [2]: https://www.cambridge.org/core/books/complexity-knots-colourings-and-countings/84DC92FA83A7B5A39231D8396321D45A
  [3]: https://en.wikipedia.org/wiki/Existential_theory_of_the_reals