On multiple roots in Descartes’ Rule and their distance to roots of higher derivatives

If an open interval I contains a k  -fold root αα of a real polynomial f, then, after transforming I   to (0,∞)(0,∞), Descartes’ Rule of Signs counts exactly k roots of f in I, provided I is such that Descartes’ Rule counts no roots of the kth 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 σσ from αα to any complex root of the k  th derivative. We relate σσ to the minimum distance s   from αα to any other complex root of f   using Szegő's composition theorem. For integer polynomials, log(1/σ)log(1/σ) obeys the same asymptotic worst-case bound as log(1/s)log(1/s).