{
  "abstract": "Let $k$ be a field of characteristic zero and $D_N(a_n)=a_{n+1}a_{n+2}$ the adjacent-quadratic locally nilpotent derivation on $A_N=k[a_0,\\dots,a_N]$. We determine the invariant ring at $N=5$, where deleting $a_0$ gives the polynomial base kernel $B^{D_5}=K=k[p,q,J_1,J_2]$, so localization gives the polynomial-line invariant algebra $(A_5^{D_5})_{pq}=K_{pq}[W]$ with $W=F_0/(p^2q^4)$, but this does not by itself determine the global kernel. An invariant $H$ of weight $24$ and $a_0$-degree $2$ lies outside $K[F_0]$, and adjoining it gives the six-generator hypersurface $C_5$; the obstruction is explained by algebraic residue behavior defeating naive denominator descent. An invariant $F_{46}^{int}$ of weight $46$ and $a_0$-degree $4$ lies outside $C_5$ and satisfies $p^2F_{46}^{int}\\in C_5$. To pass from this candidate to the full invariant ring, a second slice chart arising from $f_{13}$, together with exact $q$- and $p$-saturation, yields $A_5^{D_5}=C_5[F_{46}^{int}]$, hence finite generation, with a codimension-two complete-intersection presentation and explicit Hilbert series. At $N=6$ the transported $B$-level ring is exact, while the $A_6$ analysis is established through total degree $13$ and the full $A_6$ invariant ring remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02604",
  "author_total": 1,
  "authors": [
    {
      "name": "Ahmed M. Adly"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8p6cvrk0vva7a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02604",
  "fulltext_url": null,
  "id": "8p6cvrk0vva7a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02604",
  "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.AC"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7220392826",
  "pdf_url": "https://arxiv.org/pdf/2610.02604",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8p6cvrk0vva7a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.02604",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02604"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Invariant Rings of Adjacent-Quadratic Triangular Derivations",
  "title_text": "Invariant Rings of Adjacent-Quadratic Triangular Derivations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/8p6cvrk0vva7a",
  "venue": "arXiv",
  "versions": []
}