{
  "abstract": "Given two nonzero polynomials $f, g \\in \\mathbb{R}[x_1,\\dots,x_n]$ vanishing at the origin $0 \\in \\mathbb{R}^n$, we study the numerical range of the limit $$ \\lim_{x \\to 0} \\frac{f(x)}{g(x)}, $$ denoted by $\\mathcal{R}_{f,g}$, which is defined as the set of all possible limits of the rational function $\\displaystyle\\frac{f}{g}$ as $x$ tends to $0$. When $n = 2$ and $g$ has a non-isolated zero at $0$, we describe all possible forms of $\\mathcal{R}_{f,g}$ and provide an effective algorithm to compute this set based on Puiseux expansions. This is a follow-up to our recent results~\\cite{Dinh2025}, in which we only considered the case when $0$ is an isolated zero of $g$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36842",
  "author_total": 4,
  "authors": [
    {
      "name": "Si Tiep Dinh"
    },
    {
      "name": "Feng Guo"
    },
    {
      "name": "Hong Duc Nguyen"
    },
    {
      "name": "Tien Son Pham"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d3wap7h2e933m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36842",
  "fulltext_url": null,
  "id": "d3wap7h2e933m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36842",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7224108865",
  "pdf_url": "https://arxiv.org/pdf/2610.36842",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d3wap7h2e933m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36842",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36842"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The numerical range of limits of real rational functions",
  "title_text": "The numerical range of limits of real rational functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/d3wap7h2e933m",
  "venue": "arXiv",
  "versions": []
}