On the zeros of the derivatives of certain polynomials

摘要

We show that the derivative of the minimal polynomial of a Salem (resp. Pisot) number $alpha$ of degree $d$ (resp.~$d geq 2$) has $d-2$ zeros with modulus less than $1$ and a real zero $theta 1$ satisfying $vert theta -(d-1)alpha /dvert 1/d$ (resp.~$vert theta -(d-1)alpha /dvert 1/d$, except when $alpha$ belongs to a set of nine explicitly listed elements).