On the Lq norm of cyclotomic Littlewood polynomials on the unit circle

Abstract

Let n be the collection of all (Littlewood) polynomials of degree n with coefficients in {−1, 1}. In this paper we prove that if (P2ν) is a sequence of cyclotomic polynomials P2ν ∈ 2ν, then for every q > 2 with some a = a(q) > 1/2 depending only on q, where The case q = 4 of the above result is due to P. Borwein, Choi and Ferguson. We also prove that if (P2ν) is a sequence of cyclotomic polynomials P2ν ∈ 2ν, then for every 0 < q < 2 with some 0 < b = b(q) < 1/2 depending only on q. Similar results are conjectured for Littlewood polynomials of odd degree. Our main tool here is the Borwein–Choi Factorization Theorem.