{
  "abstract": "Let $\\mathcal{H}_{\\mathcal{Q}}$ be the field of unary germs at $0^+$ definable in the quasianalytic o-minimal structure $\\mathbb{R}_{\\mathcal{Q}}$ of Kaiser-Rolin-Speissegger. We construct a canonical ordered differential-field embedding of $\\mathcal{H}_{\\mathcal{Q}}$ into the field of natural-support generalized Laurent series. Every unary germ has a unique such asymptotic expansion, and bounded germs have expansions with only nonnegative exponents. A germ admits a convergent generalized Laurent-series representation exactly when its canonical expansion converges; in that case, every convergent representation with well-ordered support is canonical. We also prove that $\\mathbb{R}_{\\mathcal{Q}}$ is branching in the sense of Dembner [1]. A nonresonant hyperbolic saddle with divergent Dulac series yields a bounded positive real-analytic $\\mathbb{R}_{\\mathcal{Q}}$-definable germ whose canonical generalized power series diverges. This gives a negative answer to a question of Dembner asking whether every unary germ definable in a branching structure admits a convergent generalized power-series representation. Nevertheless, every bounded unary $\\mathbb{R}_{\\mathcal{Q}}$-germ has a canonical formal generalized power-series expansion. Consequently, the convergent germs form a proper ordered differential subfield of $\\mathcal{H}_{\\mathcal{Q}}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13189",
  "author_total": 1,
  "authors": [
    {
      "name": "Mostafa Mirabi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6tmce85z8q99a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13189",
  "fulltext_url": null,
  "id": "6tmce85z8q99a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13189",
  "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.DS"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2610|Mathematical and Theoretical Analysis"
    ]
  ],
  "openalex": "W7213311292",
  "pdf_url": "https://arxiv.org/pdf/2609.13189",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6tmce85z8q99a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2609.13189",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13189"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Canonical expansions of $\\mathbb R_{\\mathcal Q}$-germs",
  "title_text": "Canonical expansions of ℝ_𝒬-germs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/6tmce85z8q99a",
  "venue": "arXiv",
  "versions": []
}