Borwein and Choi conjectured that a polynomial $P(x)$ with coefficients $\pm1$ of degree $N-1$ is cyclotomic iff $$P(x)=\pm \Phi_{ p_1}(\pm x)\Phi_{ p_2}(\pm x^{ p_1})\cdots \Phi_{ p_r}(\pm x^{ p_1p_2\cdots p_{ r-1}})$$ where $N=p_1p_2\cdots p_{ r}$ and the $p_i$ are primes, not necessarily distinct. Here $\Phi_p(x):=(x^p-1)/(x-1)$ is the $p-$th cyclotomic polynomial. In \cite{ 1}, 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.
Categories: math.NT