Multiple Roots of [−1, 1] Power Series

Abstract

We are interested in how small a root of multiplicity k can be for a power series of the form f ( z ) : = 1 + ∑ n = 1 ∞ a i z i with coefficients ai in [−1, 1]. Let r(k) denote the size of the smallest root of multiplicity k possible for such a power series. We show that 1 - log ( e k 1 / 2 ) k + 1 ⩽ r ( k ) ⩽ 1 - 1 k + 1 We describe the form that the extremal power series must take and develop an algorithm that lets us compute the optimal root (which proves to be an algebraic number). The computations, for k⩽27, suggest that the upper bound is close to optimal and that r(k)∼1−c/(k+1), where c=1.230….