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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      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
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: power series, restricted coefficients, multiple roots, Lehmer's problem
      Subjects: 30-xx Functions of a complex variable > 30Cxx Geometric function theory
      12-xx Field theory and polynomials > 12Dxx Real and complex fields
      30-xx Functions of a complex variable > 30Bxx Series expansion
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 17 Nov 2003
      Last Modified: 21 Apr 2010 11:13

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