Polynesian Journal of Mathematics

Volume 4, Issue 7
Special volume AJAX in honor of René Schoof

On the convexity of the fundamental domain of binary cyclotomic forms

Étienne Fouvry and Michel Waldschmidt

Abstract:

Denote by ${\mathrm{\Phi }}_m(X,Y)\in \mathbb{Z}[X,Y]$, $m\ge 1$ the sequence of binary cyclotomic forms, so that ${\mathrm{\Phi }}_m(T,1)\in \mathbb{Z}\left[T\right]$, $m\ge 1$ is the sequence of cyclotomic polynomials. We prove that, for $m\ge 3$, the fundamental domain

${𝒪}_m:=\{(x,y)\in {ℝ}^2\mid {\mathrm{\Phi }}_m(x,y)\le 1\}$

of ${\mathrm{\Phi }}_m$ 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 $k\ge 2$. Finally we investigate the convexity of the fundamental domains attached to some products of binary forms ${\mathrm{\Phi }}_m$.

Keywords: Cyclotomic forms, convexity.

Citation:

Étienne Fouvry and Michel Waldschmidt. On the convexity of the fundamental domain of binary cyclotomic forms. Polynesian Journal of Mathematics, Volume 4, Issue 7, Pages 1–14. (Sept. 2026) DOI: 10.69763/polyjmath.4.7

Export to BibTeX

Milestones:

Received 30 Jan 2026
Revised 6 Jul 2026
Accepted 7 Jul 2026
Published 28 Sep 2026
Communicated by Samuele Anni

The above content is published under the terms of the Creative Commons Attribution 4.0 International license.