# Roots on a Ray: Power Series with Restricted Coefficients and a Root on a Given Ray

Beaucoup, Franck and Borwein, Peter and Boyd, David W. and Pinner, Christopher (1996) Roots on a Ray: Power Series with Restricted Coefficients and a Root on a Given Ray. [Preprint]

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Postscript
We consider bounds on the smallest possible root with a specified argument $\phi$ of a power series $f(z)=1+{ \sum_{n=1}^{\infty}} a_{i}z^{i}$ with coefficients $a_{i}$ in the interval [-g,g]. We describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when $\phi/2\pi$ is rational. When $g\geq 2\sqrt{2}+3$ we show that the smallest disc containing two roots has radius $(\sqrt{g}+1)^{-1}$ coinciding with the smallest double real root possible for such a series. It is clear from our computations that the behaviour is more complicated for smaller $g$. We give a similar procedure for computing the smallest circle with a real root and a pair of conjugate roots of a given argument. We conclude by briefly discussing variants of the beta-numbers (where the defining integer sequence is generated by taking the nearest integer rather than the integer part). We show that the conjugates, $\lambda$, of these pseudo-beta-numbers either lie inside the unit circle or their reciprocals must be roots of $[-1/2,1/2)$ power series; in particular we obtain the sharp inequality $|\lambda |\leq 3/2$.