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

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 $r(1/3,2;[-1,0])=.8656332072...$ .