On Multiple Roots in Descartes' Rule and Their Distance to Roots of Higher Derivatives

If an open interval $I$ contains a $k$-fold root $\alpha$ of a real polynomial~$f$, then, after transforming $I$ to $(0,\infty)$, Descartes' Rule of Signs counts exactly $k$ roots of $f$ in~$I$, provided $I$ is such that Descartes' Rule counts no roots of the $k$-th derivative of~$f$. We give a simple proof using the Bernstein basis. The above condition on $I$ holds if its width does not exceed the minimum distance $\sigma$ from $\alpha$ to any complex root of the $k$-th derivative. We relate $\sigma$ to the minimum distance $s$ from $\alpha$ to any other complex root of $f$ using Szeg{\H o}'s composition theorem. For integer polynomials, $\log(1/\sigma)$ obeys the same asymptotic worst-case bound as $\log(1/s)$