On cyclotomic polynomials with {$\pm1$} coefficients

Abstract

We characterize all cyclotomic polynomials of even degree with coefficients restricted to the set $\{+1,-1\}$. In this context a cyclotomic polynomial is any monic polynomial with integer coefficients and all roots of modulus 1. \textit{Inter alia} we characterize all cyclotomic polynomials with odd coefficients.

The characterization is as follows. A polynomial $P(x)$ with coefficients $\pm1$ of even degree $N{-}1$ is cyclotomic if and only if

$$% P(x)=\pm\funnyPhi_{p_1}(\pm x)@\funnyPhi_{p_2}(\pm x^{p_1})\cdots \funnyPhi_{p_r}(\pm x^{p_1p_2\cdots p_{r-1}}) $$%,

where $N=p_1p_2\cdots p_r$ and the $p_i$ are primes, not necessarily distinct, and where $\funnyPhi_{p}(x):= (x^p-1)/(x-1)$ is the $p$-th cyclotomic polynomial.

We conjecture that this characterization also holds for polynomials of odd degree with $\pm 1$ coefficients. This conjecture is based on substantial computation plus a number of special cases.

Central to this paper is a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials.