Fractalized cyclotomic polynomials

Abstract

For each prime power p m p^m , we realize the classical cyclotomic polynomial Φ p m ( x ) \Phi _{p^m}(x) as one of a collection of 3 m 3^m different polynomials in Z [ x ] \mathbf {Z}[x] . We show that the new polynomials are similar to Φ p m ( x ) \Phi _{p^m}(x) in many ways, including that their discriminants all have the form ± p c \pm p^c . We show also that the new polynomials are more complicated than Φ p m ( x ) \Phi _{p^m}(x) in other ways, including that their complex roots are generally fractal in appearance.