>***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 $N$ disjoint regions.

Question
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What is the expected value of $N$ (over all possible realizations of $W$) ?

Observations
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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.