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An extremal property of Fekete polynomials

Authors: Peter Borwein, Kwok-Kwong Stephen Choi and Soroosh Yazdani
Journal: Proc. Amer. Math. Soc. 129 (2001), 19-27
MSC (2000): Primary 11J54, 11B83
Published electronically: July 21, 2000
MathSciNet review: 1784013
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Abstract | References | Similar Articles | Additional Information


The Fekete polynomials are defined as

\begin{displaymath}F_q(z) := \sum^{q-1}_{k=1} \left(\frac{k}{q}\right) z^k\end{displaymath}

where $\left(\frac{\cdot}{q}\right)$ is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they provide sequences with smallest known $L_4$ norm out of the polynomials with $\pm 1$ coefficients.

The main purpose of this paper is to prove the following extremal property that characterizes the Fekete polynomials by their size at roots of unity.

Theorem 0.1. Let $f(x)=a_1x+a_2x^2+\cdots +a_{N-1}x^{N-1}$ with odd $N$ and $a_n=\pm 1$. If

\begin{displaymath}\operatorname{max}\{ \vert f(\omega^k)\vert : 0 \le k \le N-1 \} = \sqrt{N}, \end{displaymath}

then $N$ must be an odd prime and $f(x)$ is $\pm F_q(x)$. Here $ \omega:=e^{\frac{2\pi i}{N}}.$

This result also gives a partial answer to a problem of Harvey Cohn on character sums.

References [Enhancements On Off] (What's this?)

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Additional Information

Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Kwok-Kwong Stephen Choi
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, SAR, China

Soroosh Yazdani
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Keywords: Class number, $\pm1$ coefficients, merit factor, Fekete polynomials, Turyn polynomials, Littlewood polynomials
Received by editor(s): March 15, 1999
Published electronically: July 21, 2000
Additional Notes: The research of P. Borwein is supported, in part, by NSERC of Canada. K.K. Choi is a Pacific Institute of Mathematics Postdoctoral Fellow and the Institute’s support is gratefully acknowledged.
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2000 Copyright held by the authors