On the zeros of the derivatives of certain polynomials
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 \(5\)).
scientific article; zbMATH DE number 7602920