# Double Roots of $[-1,1]$ Power Series and Related Matters

Pinner, Christopher (1996) Double Roots of $[-1,1]$ Power Series and Related Matters. [Preprint]

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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 $f(x)=1+\sum_{n=1}^{\infty} a_{i}x^{i}$ with coefficients $a_{i}$ 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 $\left(\sum_{i=1}^{t} \delta (\theta_{i})k_{i}\right) -1$ 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 $r(1/3,2;[-1,0])=.8656332072...$ .