PreprintCC BY 4.0arXiv
Unimodular Zeros of Selmer's Harmonic Trinomials
Abstract
Let n>k be positive integers and let ε,δ∈{−1,1}. In this note, we classify the unimodular zeros of the harmonic trinomial Hε,δ:ℂ→ ℂ given by Hε,δ(z)=zn+εz̅ k+δ, where z̅ is the complex conjugate of z∈ ℂ. Let g:=gcd(n,k) be the greatest common divisor of n and k, and set n0:=n/g and k0:=k/g. For each choice of ε,δ∈ {−1,1}, we give a necessary and sufficient congruence involving n0 and k0 to guarantee that Hε,δ has unimodular zeros. More precisely, if the corresponding congruence holds, then Hε,δ has exactly 2g unimodular zeros, which are given explicitly; otherwise, it has no unimodular zeros. The proof uses a reduction to a trinomial equation and elementary divisibility arguments.
Related
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- Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height $1$
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- On the index of generalized trinomials over Valued fields
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BibTeX
@misc{barrera2026unimodular,
title = {{Unimodular Zeros of Selmer's Harmonic Trinomials}},
author = {Gerardo Barrera and Waldemar Barrera and Juan Pablo Navarrete},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22132},
eprint = {2610.22132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22132},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unimodular Zeros of Selmer's Harmonic Trinomials
AU - Gerardo Barrera
AU - Waldemar Barrera
AU - Juan Pablo Navarrete
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22132
UR - https://arxiv.org/abs/2610.22132
AB - Let n>k be positive integers and let ε,δ∈{−1,1}. In this note, we classify the unimodular zeros of the harmonic trinomial H_ε,δ:ℂ→ ℂ given by H_ε,δ(z)=z^n+εz̅^ k+δ, where z̅ is the complex conjugate of z∈ ℂ. Let g:=gcd(n,k) be the greatest common divisor of n and k, and set n_0:=n/g and k_0:=k/g. For each choice of ε,δ∈ {−1,1}, we give a necessary and sufficient congruence involving n_0 and k_0 to guarantee that H_ε,δ has unimodular zeros. More precisely, if the corresponding congruence holds, then H_ε,δ has exactly 2g unimodular zeros, which are given explicitly; otherwise, it has no unimodular zeros. The proof uses a reduction to a trinomial equation and elementary divisibility arguments.
ER -