On the convexity of the fundamental domain of binary cyclotomic forms

Abstract

Denote by ${\Phi }_m(X,Y)\in \mathbb{Z}[X,Y]$, $m⩾1$ the sequence of binary cyclotomic forms, so that ${\Phi }_m(T,1)\in \mathbb{Z}[T]$, $m⩾1$ is the sequence of cyclotomic polynomials. We prove that, for $m⩾3$, the fundamental domain ${\mathcal{O}}_m:=\{(x,y)\in {\mathbb{R}}^2\mid {\Phi }_m(x,y)⩽1\}$ of ${\Phi }_m$ is a convex subset of ${\mathbb{R}}^2$ if and only if $m$ is equal to $p$, $2p$ or $2^k$ with $p$ an odd prime and $k⩾2$. Finally we investigate the convexity of the fundamental domains attached to some products of binary forms ${\Phi }_m$.