An extremal property of Fekete polynomials

• Published in: Proceedings of the American Mathematical Society 2001-01, Vol.129 (1), p.19-27
• Main Authors: Borwein, Peter, Choi, Kwok-Kwong Stephen, Yazdani, Soroosh
• Format: Article
• Language: English
• Subjects: Coefficients; Combinatorics; Integers; Mathematical theorems; Number theory; Polynomials; Property titles; Symbols; Yazdanism
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-00-05798-1

The Fekete polynomials are defined as \[F_q(z) := \sum^{q-1}_{k=1} \left(\frac{k}{q}\right) z^k\] where \left(\frac{·}{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. \vspace{6pt} \noindent{\bf Theorem 0.1.} {\em Let f(x)=a_1x+a_2x^2+\cdots +a_{N-1}x^{N-1} with odd N and a_n=\pm 1. If \[ \operatorname{max}{ |f(\omega^k)| : 0 \le k \le N-1 } = \sqrt{N}, \] then N must be an odd prime and f(x) is \pm F_q(x). Here \omega:=e^{\frac{2\pi i}{N}}.} \vspace{6pt} This result also gives a partial answer to a problem of Harvey Cohn on character sums.