A number theoretic problem on the distribution of polynomials with bounded roots

Let ${\mathcal{E}}_d^{(s)}$ denote the set of coefficient vectors $(a_1,\dots ,a_d)\in {\mathbb{R}}^d$ of contractive polynomials $x^d+a_1{x}^{d-1}+\cdots +a_d\in \mathbb{R}[x]$ that have exactly $s$ pairs of complex conjugate roots and let $v_d^{(s)}={\lambda }_d({\mathcal{E}}_d^{(s)})$ be its ($d$-dimensional) Lebesgue measure. We settle the instance $s=1$ of a conjecture by Akiyama and Peth\H{o}, stating that the ratio $v_d^{(s)}\mathrm{/}{v}_d^{(0)}$ is an integer for all $d\ge 2s\mathrm{.}$ Moreover we establish the surprisingly simple formula $v_d^{(1)}\mathrm{/}{v}_d^{(0)}=(P_d(3)-2d-1)\mathrm{/}4,$ where $P_d(x)$ are the Legendre polynomials.