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

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 of a power series with coefficients in the interval . We describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when is rational. When we show that the smallest disc containing two roots has radius 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 . 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, , of these pseudo-beta-numbers either lie inside the unit circle or their reciprocals must be roots of power series; in particular we obtain the sharp inequality .

**1.**F. Beaucoup, P. Borwein, D. W. Boyd and C. Pinner, Multiple roots of ,*J. London Math. Soc. to appear*.**2.**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****3.**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**, 10.1112/plms/s3-68.3.477**4.**Osami Yamamoto,*On some bounds for zeros of norm-bounded polynomials*, J. Symbolic Comput.**18**(1994), no. 5, 403–427. MR**1327384**, 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

Email:
pinner@cecm.sfu.ca

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

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