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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Polynomials with coefficients from a finite set


Authors: Peter Borwein, Tamás Erdélyi and Friedrich Littmann
Journal: Trans. Amer. Math. Soc. 360 (2008), 5145-5154
MSC (2000): Primary 30B30; Secondary 11C08, 30C15
Published electronically: May 27, 2008
MathSciNet review: 2415068
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Abstract | References | Similar Articles | Additional Information

Abstract: In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of $ \mathbb{C}$ is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem.

This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of $ \mathbb{C}$ cannot tend to zero uniformly on an arc of the unit circle.

Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let $ n$ be so large that

$\displaystyle \delta_n:=33\pi \frac{\log n}{\sqrt{n}} $

satisfies $ \delta_n\le 1$. We show that any polynomial in

$\displaystyle K_n$ $\displaystyle :=\Big\{\sum_{k=0}^n a_k z^k\,:\,\vert a_0\vert=\vert a_n\vert=1$ and $\displaystyle \vert a_k\vert\le 1\Big\}$    

has at least

$\displaystyle 8\sqrt{n}\log n $

zeros in any disk with center on the unit circle and radius $ \delta_n$.


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Additional Information

Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: pborwein@cecm.sfu.ca

Tamás Erdélyi
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: terdelyi@math.tamu.edu

Friedrich Littmann
Affiliation: Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105
Email: Friedrich.Littmann@ndsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04605-9
PII: S 0002-9947(08)04605-9
Keywords: Zeros, rational functions, Duffin--Schaeffer Theorem, Littlewood polynomials
Received by editor(s): June 8, 2005
Received by editor(s) in revised form: February 15, 2006
Published electronically: May 27, 2008
Article copyright: © Copyright 2008 by the authors