On the zeros of the derivatives of certain polynomials (Q2084270)

In this paper the author obtains some bounds on the zeros of the derivative \(P'\), where \(P\) is the minimal polynomial of a degree \(d\) Salem number or a Pisot number. In particular, he shows that if \(\alpha\) is a Salem number of degree \(d\) then the derivative \(P'\) of the minimal of polynomial \(P\) of \(\alpha\) has a unique root \(\theta\) outside the unit circle. For this root \(\theta\), the following inequalities hold: \[ \max\Big\{\frac{\alpha+\alpha^{-1}}{2}, \frac{(d-1)\alpha-1}{d}\Big\} <\theta<\frac{(d-1)\alpha+1}{d}. \] For Pisot number \(\alpha\) the situation is similar. The derivative \(P'\) has a unique root \(\theta\) outside the unit circle. It satisfies \(|\theta-(d-1)\alpha/d|<1/d\) except when the minimal polynomial \(P\) of \(\alpha\) is one of \(9\) explicitly listed polynomials (of degrees at most \(5\)).