On the convexity of the fundamental domain of binary cyclotomic forms

arXiv:2610.37735v1 Announce Type: new Abstract: Denote by $\Phi_m(X,Y)\in\Z[X,Y]$, $m\geqslant 1$ the sequence of binary cyclotomic forms, so that $\Phi_m(T,1)\in\Z[T]$, $m\geqslant 1$ is the sequence of cyclotomic polynomials. We prove that, for $m\geqslant 3$, the fundamental domain $ \cal{O}_m:= \{ (x,y)\in\R^2\; \mid \; \Phi_m(x,y)\leqslant 1\} $ of $\Phi_m$ is a convex subset of $\R^2$ if and only if $m$ is equal to $p$, $2p$ or $2^k$ with $p$ an odd prime and $k\geqslant 2$. Finally we investigate the convexity of the fundamental domains attached to some products of binary forms $\Phi_m$.

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