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

摘要

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.The above 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 Szegő's composition theorem. For integer polynomials, log(1/σ) obeys the same asymptotic worst-case bound as log(1/s).