# Multiple Roots of [-1,1] Power Series

Beaucoup, Franck and Borwein, Peter and Boyd, David W. and Pinner, Christopher (1995) Multiple Roots of [-1,1] Power Series. [Preprint]

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## Abstract

We are interested in how small a root of multiplicity $k$ can be for a power series of the form $f(z):=1+{\sum_{n=1}^{\infty}} a_{i}z^{i}$ with coefficients $a_{i}$ 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 $1-\frac{\log (e\sqrt{k})}{k+1} \leq r(k) \leq 1 -\frac{1}{k+1}.$ 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 $k\leq 27$, suggest that the upper bound is close to optimal and that $r(k)\sim 1 -c/(k+1)$ where $c=1.230...$.

Item Type: Preprint pubdom FALSE power series, restricted coefficients, multiple roots, Lehmer's problem 30-xx Functions of a complex variable > 30Cxx Geometric function theory12-xx Field theory and polynomials > 12Dxx Real and complex fields30-xx Functions of a complex variable > 30Bxx Series expansion UNSPECIFIED Users 1 not found. 17 Nov 2003 21 Apr 2010 11:13 https://docserver.carma.newcastle.edu.au/id/eprint/120