Power series with restricted coefficients and a root on a given ray

Authors:
Franck Beaucoup, Peter Borwein, David W. Boyd and Christopher Pinner

Journal:
Math. Comp. **67** (1998), 715-736

MSC (1991):
Primary 30C15; Secondary 30B10, 12D10

DOI:
https://doi.org/10.1090/S0025-5718-98-00960-0

MathSciNet review:
1468939

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Abstract | References | Similar Articles | Additional Information

Abstract: 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 $|\lambda |\leq 3/2$.

- F. Beaucoup, P. Borwein, D. W. Boyd and C. Pinner, Multiple roots of $[-1,1]$,
*J. London Math. Soc. to appear*. - A. M. Odlyzko and B. Poonen,
*Zeros of polynomials with $0,1$ coefficients*, Enseign. Math. (2)**39**(1993), no. 3-4, 317–348. MR**1252071** - Boris Solomyak,
*Conjugates of beta-numbers and the zero-free domain for a class of analytic functions*, Proc. London Math. Soc. (3)**68**(1994), no. 3, 477–498. MR**1262305**, DOI https://doi.org/10.1112/plms/s3-68.3.477 - Osami Yamamoto,
*On some bounds for zeros of norm-bounded polynomials*, J. Symbolic Comput.**18**(1994), no. 5, 403–427. MR**1327384**, DOI https://doi.org/10.1006/jsco.1994.1056

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Additional Information

**Franck Beaucoup**

Affiliation:
Equipe de Mathématiques appliquées, Ecole des Mines de Saint-Etienne, 42023 Saint-Etienne, France

Email:
beaucoup@emse.fr

**Peter Borwein**

Affiliation:
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada

Email:
pborwein@cecm.sfu.ca

**David W. Boyd**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

Email:
boyd@math.ubc.ca

**Christopher Pinner**

Affiliation:
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada & Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

MR Author ID:
319822

Email:
pinner@cecm.sfu.ca

Keywords:
Power series,
restricted coefficients,
beta-numbers

Received by editor(s):
July 15, 1996

Additional Notes:
Research of the second and third authors was supported by the NSERC

Article copyright:
© Copyright 1998
by the authors