Context. This is the first in a set of tiny pieces of a problem I've formulated to help me measure the "complexity" of certain piecewise linear functions. Thanks in advance for your help and patience.
So, let $W$ be drawn from a Ginibre ensemble (i.e the entries are i.i.d $\mathcal N(0, 1)$) of size $n\times m$ and let $b \in \mathbb R^n$ be fixed (we might consider variations where $b$ is random...). For $1 \le i \le n$, consider the random hyperplane defined by $\mathcal H_i := \{x \in \mathbb R^m \mid w_i^Tx = b_i\}$. The $\mathcal H_i$'s cut $\mathbb R^m$ into $N_W \ge 1$ disjoint regions.
Question
What is the expected value of $N_W$ over all possible realizations of $W$ ?
Observations
Somehow, I feel I should be able to pull off something via the Kac-Rice formula for random Gaussian fields, but I don't quite know how to start.
It's not difficult to get the upper bound $N_W \le \sum_{j=0}^{\text{rank}(W)} {{n}\choose{j}}$. For example, one can use the counting arguments in https://math.stackexchange.com/a/339853/168758. Thus, using the fact $W$ is of full rank $r=\max(n,m)$ almost surely, we have the (very very crude bound) $$ \begin{split} \mathbb E_W[N_W] \le \mathbb E_W\left[\sum_{j=0}^{\text{rank}(W)} {{n}\choose{j}}\mid \text{rank}(W)=k\right] &= \sum_{k=0}^{r} P(\text{rank}(W)=k)\sum_{j=0}^k {{n}\choose{j}}\\ &= \sum_{j=0}^r {{n}\choose{j}} \end{split} $$
