Flat Cyclotomic Polynomials: A New Approach

Abstract

We build a new theory for analyzing the coefficients of any cyclotomic polynomial by considering it as a gcd of simpler polynomials. Using this theory, we generalize a result known as periodicity and provide two new families of flat cyclotomic polynomials. One, of order 3, was conjectured by Broadhurst: ${\Phi }_{pqr}(x)$ is flat where $p<q<r$ are primes and there is a positive integer $w$ such that $r\equiv \pm w(\mathrm{mod}pq)$, $p\equiv 1(\mathrm{mod}w)$ and $q\equiv 1(\mathrm{mod}wp)$. The other is the first general family of order 4: ${\Phi }_{pqrs}(x)$ is flat for primes $p,q,r,s$ where $q\equiv -1(\mathrm{mod}p)$, $r\equiv \pm 1(\mathrm{mod}pq)$, and $s\equiv \pm 1(\mathrm{mod}pqr)$. Finally, we prove that the natural extension of this second family to order 5 is never flat, suggesting that there are no flat cyclotomic polynomials of order 5.

52 pages; to be submitted to International Journal of Number Theory