Fractalized cyclotomic polynomials

Abstract:

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