Peer-reviewedCC BY 4.0Polynesian Journal of Mathematics
On the convexity of the fundamental domain of binary cyclotomic forms
Abstract
Denote by Φm(X,Y)∈Z[X,Y], mgeqslant 1 the sequence of binary cyclotomic forms, so that Φm(T,1)∈Z[T], mgeqslant 1 is the sequence of cyclotomic polynomials. We prove that, for mgeqslant 3, the fundamental domain calOm:= { (x,y)∈R2 | Φm(x,y)leqslant 1} of Φm is a convex subset of R2 if and only if m is equal to p, 2p or 2k with p an odd prime and kgeqslant 2. Finally we investigate the convexity of the fundamental domains attached to some products of binary forms Φm.
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BibTeX
@article{fouvry2026convexity,
title = {{On the convexity of the fundamental domain of binary cyclotomic forms}},
author = {Étienne Fouvry and Michel Waldschmidt},
journal = {Polynesian Journal of Mathematics},
year = {2026},
month = sep,
doi = {10.69763/polyjmath.4.7},
eprint = {2610.37735},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05837700v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - On the convexity of the fundamental domain of binary cyclotomic forms
AU - Étienne Fouvry
AU - Michel Waldschmidt
PY - 2026
DA - 2026/09/28
JO - Polynesian Journal of Mathematics
DO - 10.69763/polyjmath.4.7
UR - https://hal.science/hal-05837700v1
AB - Denote by Φ_m(X,Y)∈Z[X,Y], mgeqslant 1 the sequence of binary cyclotomic forms, so that Φ_m(T,1)∈Z[T], mgeqslant 1 is the sequence of cyclotomic polynomials. We prove that, for mgeqslant 3, the fundamental domain calO_m:= { (x,y)∈R^2 | Φ_m(x,y)leqslant 1} of Φ_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 kgeqslant 2. Finally we investigate the convexity of the fundamental domains attached to some products of binary forms Φ_m.
SN - 3075-3422
SN - 3075-3422
ER -