{
  "abstract": "Let $E/K$ be an elliptic curve over a field defined by the short Weierstrass equation and let $\\mathbf{P}=(P_1, P_2) \\in E(K)^2\\setminus \\{(O, O), (\\pm P, \\pm P)\\}$ be a pair of non-torsion points, where $O$ denotes the point at infinity. We give a uniform upper bound for the quantity $\\min(ν(Φ_{(v_1, v_2)}(\\mathbf{P})), 2ν(Ψ_{(v_1, v_2)}(\\mathbf{P})))$, where $ν$ is a normalised discrete valuation associated to primes where $E$ has bad reduction and $Ψ^2_{(v_1, v_2)}(\\mathbf{P})$, $Φ_{(v_1, v_2)}(\\mathbf{P})$ are rational functions for describing $\\mathbb{Z}$--linear combination of points $[v_1]P_1+[v_2]P_2$ on $E$, higher--dimensional analogues of division polynomials $ψ^2_n, ϕ_n$ known as net polynomials. They are also elliptic nets, which are $n$--dimensional arrays with values in $K$ satisfying the same nonlinear recurrence relation that division polynomials do as well. Hence, they are useful in studying counting problems on elliptic curves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20273",
  "author_total": 1,
  "authors": [
    {
      "name": "Hang Lok Edison Au-Yeung"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5e64bthg2fjk7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20273",
  "fulltext_url": null,
  "id": "5e64bthg2fjk7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20273",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7222066334",
  "pdf_url": "https://arxiv.org/pdf/2610.20273",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5e64bthg2fjk7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20273",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20273"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "An upper bound for valuations of elliptic nets at primes of bad reduction",
  "title_text": "An upper bound for valuations of elliptic nets at primes of bad reduction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/5e64bthg2fjk7",
  "venue": "arXiv",
  "versions": []
}