An extremal property of fekete polynomials

Abstract

The Fekete polynomials are defined as [GRAPHICS] where (./q) 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 +/-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(1)x + a(2)x (2) + ... + a(N-1) x(N-1) with odd N and a(n) = +/-1. If [GRAPHICS] then N must be an odd prime and f(x) is +/- F-q (x). Here w:= e 2 pi i/N. This result also gives a partial answer to a problem of Harvey Cohn on character sums.