{
  "abstract": "The Kantowski--Sachs interior of Grumiller's two-dimensional dilaton gravity model reduces to $\\dot H=\\tfrac12(Λ-3H^2-u^2)+Qu$, $\\dot u=-Hu$, where $H$ is the expansion rate of the orbit two-spheres, $u$ their inverse areal radius, and $Q$ is minus twice the Rindler acceleration. For $Q\\neq0$ it has two distinct rank-one nilpotent equilibria with complementary Bogdanov--Takens (BT) degeneracies: $(a,b)=(0,\\neq0)$ with multiplicity $μ=3$, and $(a,b)=(\\neq0,0)$ with $μ=2$. We show that this complementarity is forced by a single divisor-organized structure. The system is Darboux integrable, $X=\\tfrac12u^4X_I$, where $I$ is the mass function and $u^4$ is an inverse integrating factor whose zero divisor is the invariant axis $\\{u=0\\}$. Off the divisor, every nilpotent equilibrium of $RX_I$ with $R(p)\\neq0$ has $b=0$, from $b=D_{q_0}\\operatorname{tr}DX$. On the divisor, invariance of the coordinate axis forces $a=0$ because $\\det DX|_{u=0}=Φ_xG$ and both factors vanish at the corner. Hence $ab=0$ throughout the class, so a versal two-parameter BT unfolding is impossible. The Bernstein--Kushnirenko bound fails at both points; instead, the exact local identity $μ_0=m_1\\ell_y+m_2\\ell_x$ gives $μ=3,2$ without a nondegeneracy hypothesis. Both multiplicities have a mixed-covolume interpretation, with $\\operatorname{Covol}=\\sum_{k,i}\\min(p_kq_i',p_i'q_k)$ on convenient diagrams; the corresponding general theorem for the non-convenient diagrams arising here remains open. For $\\dot H=\\tfrac{1-m}{2}H^2+ψ(u)$, $\\dot u=-Hu$, with $ψ(0)=0$, $ψ'(0)\\neq0$, $m\\neq1$, the vacuum always has $μ=3$, $a=0$, $b=-m$. Its Dumortier--Llibre--Artés discriminant is $b_1^2+8a_3=(m-2)^2$, so for every $m>1$ it has one hyperbolic and one elliptic sector, independently of $ψ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16677",
  "author_total": 3,
  "authors": [
    {
      "name": "E. Chan-López"
    },
    {
      "name": "A. Martín-Ruiz"
    },
    {
      "name": "J. M. Paulin Fuentes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1z7apwwva9579.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16677",
  "fulltext_url": null,
  "id": "1z7apwwva9579",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16677",
  "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-ph"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7213421118",
  "pdf_url": "https://arxiv.org/pdf/2609.16677",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1z7apwwva9579.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:41Z",
      "native_id": "2609.16677",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16677"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Newton geometry of Bogdanov-Takens degeneracies in integrable dilatonic models: invariant divisors and boundary multiplicity",
  "title_text": "Newton geometry of Bogdanov-Takens degeneracies in integrable dilatonic models: invariant divisors and boundary multiplicity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/1z7apwwva9579",
  "venue": "arXiv",
  "versions": []
}