Notes on the Borwein-Choi conjecture of Littlewood cyclotomic polynomials

Abstract

Borwein and Choi conjectured that a polynomial P(x) with coefficients ±1 of degree N − 1 is cyclotomic iff $$ P(x) = \pm \Phi _{p_1 } ( \pm x)\Phi _{p_2 } ( \pm x^{p_1 } ) \cdots \Phi _{p_r } ( \pm x^{p_1 p_2 \cdots p_{r - 1} } ), $$ , where N = p 1 p 2 … p r and the p i are primes, not necessarily distinct. Here Φ p (x):= (x p − 1)/(x − 1) is the p-th cyclotomic polynomial. They also proved the conjecture for N odd or a power of 2. In this paper we introduce a so-called E-transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the conjecture.