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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 $X^d+a_1X^{d-1}+\cdots+a_{d-1}X+a_d$ whoseroots all lie in the open unit disk. This region has been extensively studiedover decades. Recently, Akiyama and Peth\H{o} considered the subsets$\mathcal{E}_d^{(s)}$ of the Schur-Cohn region that correspond to polynomialsof degree $d$ with exactly $s$ pairs of nonreal roots. They were especiallyinterested in the $d$-dimensional Lebesgue measures$v_d^{(s)}:=\lambda_d(\mathcal{E}_d^{(s)})$ of these sets and their arithmeticproperties, and gave some fundamental results. Moreover, they posed twoconjectures that we prove in the present paper. Namely, we show that in thetotally 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 generalcase 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 explicitformul\ae{} for $v_d^{(s)} / v_d^{(0)}$ for arbitrary $d\in \mathbb{N}$, $s\led/2$. The ingredients of our proofs comprise Selberg type integrals,determinants like the Cauchy double alternant, and partial Hilbert matrices.

Distribution results on polynomials with bounded roots