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

If an open interval I contains a k-fold root α of a real polynomial f, then, after transforming I to ( 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. ove condition on I holds if its width does not exceed the minimum distance σ from α to any complex root of the kth derivative. We relate σ to the minimum distance s from α to any other complex root of f using Szegőʹs composition theorem. For integer polynomials, log ( 1 / σ ) obeys the same asymptotic worst-case bound as log ( 1 / s ) .