Short Communication: On multiple roots in Descartes' Rule and their distance to roots of higher derivatives

If an open interval I contains a k-fold root @a 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 @s from @a to any complex root of the kth derivative. We relate @s to the minimum distance s from @a to any other complex root of f using Szego's composition theorem. For integer polynomials, log(1/@s) obeys the same asymptotic worst-case bound as log(1/s).