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Mathematics of Computation

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Double roots of $[-1,1]$ power series
and related matters

Author: Christopher Pinner
Journal: Math. Comp. 68 (1999), 1149-1178
MSC (1991): Primary 30C15; Secondary 30B10, 12D10
Published electronically: February 10, 1999
MathSciNet review: 1620243
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Abstract | References | Similar Articles | Additional Information

Abstract: For a given collection of distinct arguments $\vec{\theta}=(\theta _{1},\ldots ,\theta _{t})$, multiplicities $\vec{k}=(k_{1},\ldots ,k_{t}),$ and a real interval $I=[U,V]$ containing zero, we are interested in determining the smallest $r$ for which there is a power series $f(x)=1+\sum _{n=1}^{\infty} a_{n}x^{n}$ with coefficients $a_{n}$ in $I$, and roots $\alpha _{1}=re^{2\pi i\theta _{1}}, \ldots ,\alpha _{t}=re^{2\pi i\theta _{t}}$ of order $k_{1},\ldots ,k_{t}$ respectively. We denote this by $r(\vec{\theta},\vec{k};I)$. We describe the usual form of the extremal series (we give a sufficient condition which is also necessary when the extremal series possesses at least $ \left(\sum _{i=1}^{t} \delta (\theta _{i})k_{i}\right) -1$ non-dependent coefficients strictly inside $I$, where $\delta (\theta _{i})$ is 1 or 2 as $\alpha _{i}$ is real or complex). We focus particularly on $r(\theta,2;[-1,1])$, the size of the smallest double root of a $[-1,1]$ power series lying on a given ray (of interest in connection with the complex analogue of work of Boris Solomyak on the distribution of the random series $\sum \pm \lambda^{n}$). We computed the value of $r(\theta,2; [-1,1])$ for the rationals $\theta$ in $(0,1/2)$ of denominator less than fifty. The smallest value we encountered was $r(4/29,2;[-1,1])=0.7536065594...$. For the one-sided intervals $I=[0,1]$ and $[-1,0]$ the corresponding smallest values were $r(11/30,2;[0,1])=.8237251991... $ and $r(1/3,2;[-1,0])=.8656332072...$ .

References [Enhancements On Off] (What's this?)

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Additional Information

Christopher Pinner
Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada & Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
Address at time of publication: Mathematics and Computer Science, University of Northern British Columbia, 3333 University Way, Prince George, BC V2N 4Z9, Canada

Keywords: Power series, restricted coefficients, double roots
Received by editor(s): July 12, 1996
Received by editor(s) in revised form: November 7, 1997
Published electronically: February 10, 1999
Article copyright: © Copyright 1999 American Mathematical Society