  For $d \in \mathbb{N}$ the well-known Schur-Cohn region $\mathcal{E}_d$
consists of all $d$-dimensional vectors $(a_1,\ldots,a_d)\in\mathbb{R}^d$
corresponding to monic polynomials $X^d+a_1X^{d-1}+\cdots+a_{d-1}X+a_d$ whose
roots all lie in the open unit disk. This region has been extensively studied
over decades. Recently, Akiyama and Peth\H{o} considered the subsets
$\mathcal{E}_d^{(s)}$ of the Schur-Cohn region that correspond to polynomials
of degree $d$ with exactly $s$ pairs of nonreal roots. They were especially
interested in the $d$-dimensional Lebesgue measures
$v_d^{(s)}:=\lambda_d(\mathcal{E}_d^{(s)})$ of these sets and their arithmetic
properties, and gave some fundamental results. Moreover, they posed two
conjectures that we prove in the present paper. Namely, we show that in the
totally complex case $d=2s$ the formula \[ \frac{v_{2s}^{(s)}}{v_{2s}^{(0)}} =
2^{2s(s-1)}\binom {2s}s \] holds for all $s\in\mathbb{N}$ and in the general
case the quotient $v_d^{(s)}/v_d^{(0)}$ is an integer for all choices $d\in
\mathbb{N}$ and $s\le d/2$. We even go beyond that and prove explicit
formul\ae{} for $v_d^{(s)} / v_d^{(0)}$ for arbitrary $d\in \mathbb{N}$, $s\le
d/2$. The ingredients of our proofs comprise Selberg type integrals,
determinants like the Cauchy double alternant, and partial Hilbert matrices.
