{
  "abstract": "Let $n>k$ be positive integers and let $\\varepsilon,δ\\in\\{-1,1\\}$. In this note, we classify the unimodular zeros of the harmonic trinomial $H_{\\varepsilon,δ}:\\mathbb{C}\\to \\mathbb{C}$ given by \\[ H_{\\varepsilon,δ}(z)=z^n+\\varepsilon\\overline z^{\\,k}+δ, \\] where $\\overline{z}$ is the complex conjugate of $z\\in \\mathbb{C}$. 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 $\\varepsilon,δ\\in \\{-1,1\\}$, we give a necessary and sufficient congruence involving $n_0$ and $k_0$ to guarantee that $H_{\\varepsilon,δ}$ has unimodular zeros. More precisely, if the corresponding congruence holds, then $H_{\\varepsilon,δ}$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22132",
  "author_total": 3,
  "authors": [
    {
      "name": "Gerardo Barrera"
    },
    {
      "name": "Waldemar Barrera"
    },
    {
      "name": "Juan Pablo Navarrete"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6tvq4yr2467fj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22132",
  "fulltext_url": null,
  "id": "6tvq4yr2467fj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22132",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.22132",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6tvq4yr2467fj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22132",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22132"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Unimodular Zeros of Selmer's Harmonic Trinomials",
  "title_text": "Unimodular Zeros of Selmer's Harmonic Trinomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/6tvq4yr2467fj",
  "venue": "arXiv",
  "versions": []
}