Notes on the Borwein-Choi conjecture of Littlewood cyclotomic polynomials

Borwein and Choi conjectured that a polynomial P(x) with coefficients ±1 of degree N − 1 is cyclotomic iff , 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. Research partially supported by Program for New Century Excellent Talents in University Grant # NCET-06-0785 and by SRF for ROCS, SEM