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

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Let Ed⁽ˢ⁾ denote the set of coefficient vectors (a₁,,ad)∈ Rᵈ of contractive polynomials xᵈ+a₁xᵈ⁻¹++ad∈ R[x] that have exactly s pairs of complex conjugate roots and let vd⁽ˢ⁾=λd(Ed⁽ˢ⁾) be its (d-dimensional) Lebesgue measure. We settle the instance s=1 of a conjecture by Akiyama and Pethő, stating that the ratio vd⁽ˢ⁾/vd⁽⁰⁾ is an integer for all d≥ 2s. Moreover we establish the surprisingly simple formula vd⁽¹⁾/vd⁽⁰⁾ = (Pd(3)-2d-1)/4, where Pd(x) are the Legendre polynomials.