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MULTIPLE ROOTS OF [−1, 1] POWER SERIES

Published online by Cambridge University Press:  01 February 1998

FRANK BEAUCOUP ,
PETER BORWEIN ,
DAVID W. BOYD  and
CHRISTOPHER PINNER
FRANK BEAUCOUP
Affiliation:
Equipe de Mathématiques Appliquées, Ecole des Mines de Saint-Etienne, 42023 Saint-Etienne, France. E-mail address: [email protected]
PETER BORWEIN
Affiliation:
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada. E-mail address: [email protected]; [email protected]
DAVID W. BOYD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada. E-mail address: [email protected]
CHRISTOPHER PINNER
Affiliation:
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada. E-mail address: [email protected]; [email protected]
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Abstract

We are interested in how small a root of multiplicity k can be for a power series of the form f(z:= 1+[sum ]∞n=1aizi with coefficients ai in [−1, 1]. Let r(k) denote the size of the smallest root of multiplicity k possible for such a power series. We show that

formula here

We describe the form that the extremal power series must take and develop an algorithm that lets us compute the optimal root (which proves to be an algebraic number). The computations, for k[les ]27, suggest that the upper bound is close to optimal and that r(k)∼1−c/(k+1), where c=1.230….

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 57 , Issue 1 , February 1998 , pp. 135 - 147
Copyright
The London Mathematical Society 1998

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