Double roots of power series

and related matters

Author:
Christopher Pinner

Journal:
Math. Comp. **68** (1999), 1149-1178

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

DOI:
https://doi.org/10.1090/S0025-5718-99-01042-X

Published electronically:
February 10, 1999

MathSciNet review:
1620243

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a given collection of distinct arguments , multiplicities and a real interval containing zero, we are interested in determining the smallest for which there is a power series with coefficients in , and roots of order respectively. We denote this by . 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 non-dependent coefficients strictly inside , where is 1 or 2 as is real or complex). We focus particularly on , the size of the smallest double root of a 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 ). We computed the value of for the rationals in of denominator less than fifty. The smallest value we encountered was . For the one-sided intervals and the corresponding smallest values were and .

**1.**F. BEAUCOUP, P. BORWEIN, D. W. BOYD & C. PINNER, Multiple roots of power series,*J. London Math. Soc.*(2)**57**(1998), 135-147.**2.**F. BEAUCOUP, P. BORWEIN, D. W. BOYD & C. PINNER, Power series with restricted coefficients and a root on a given ray,*Math. Comp.***67**(1998), 715-736. CMP**98:07****3.**P. BORWEIN, T ERDÉLYI & G. KÓS, Littlewood-type problems on [0,1], Bull. London Math. Soc. to appear.**4.**A. ODLYZKO & B. POONEN, Zeros of polynomials with 0,1 coefficients,*L'Enseignement Math.***39**(1993), 317-348. MR**95b:10026****5.**B. SOLOMYAK, Conjugates of beta-numbers and the zero-free domain for a class of analytic functions,*Proc. London Math. Soc. (3)***68**(1994), 477-498. MR**95c:30010****6.**B. SOLOMYAK. On the random series (an Erdös problem),*Ann. Math.***142**(1995), 611-625. MR**97d:11125**

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Additional Information

**Christopher Pinner**

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

Address at time of publication:
Mathematics and Computer Science, University of Northern British Columbia, 3333 University Way, Prince George, BC V2N 4Z9, Canada

Email:
pinnerc@unbc.ca

DOI:
https://doi.org/10.1090/S0025-5718-99-01042-X

Keywords:
Power series,
restricted coefficients,
double roots

Received by editor(s):
July 12, 1996

Received by editor(s) in revised form:
November 7, 1997

Published electronically:
February 10, 1999

Article copyright:
© Copyright 1999
American Mathematical Society