{
  "abstract": "The Jacobian conjecture, open since 1939, asks whether every polynomial map of $\\C^n$ whose Jacobian determinant is a nonzero constant must have a polynomial inverse. It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite family by Gallagher (July 20) and a geometric explanation by Speyer (July 23): the counterexample sweeps the tangent lines of a plane curve. We give a self-contained account of this tangent-sweep mechanism and generalize it from plane curves to direction fields on hypersurfaces. The resulting construction, including stabilization of the three-dimensional family, produces counterexamples in every dimension greater than two and, in each dimension, of arbitrarily large geometric degree (the number of preimages of a typical point). We work it out in five explicit maps: one three-dimensional of degree four, two four-dimensional of degrees five and ten, and two five-dimensional of degrees six and twelve. We show that our explicit examples in dimensions $n\\ge4$ are not polynomially left-right equivalent to the counterexamples of Alpöge and Gallagher when these are embedded in $\\C^n$ by adjoining identity coordinates. The counterexamples provide explicit examples of étale morphisms $\\C^n\\to\\C^n$ that are not proper: they are everywhere unramified, and fail to be injective only through points escaping to infinity. All identities were verified in exact rational arithmetic. Part~\\ref{part:equivalence} proves the higher-dimensional inequivalence results, and Appendix A determines exact fiber structures for the three-dimensional examples. A complete classification of the fibers of $F_4,F_5,F_6,F_7$ remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.00222",
  "author_total": 1,
  "authors": [
    {
      "name": "Shuhong Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d4332mqram8mr.bib",
  "cited_by": 13,
  "doi": "10.48550/arxiv.2608.00222",
  "fulltext_url": null,
  "id": "d4332mqram8mr",
  "issns": [
    "2331-8422"
  ],
  "landing_url": "https://arxiv.org/abs/2608.00222",
  "language": "en",
  "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.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7172413910",
  "pdf_url": "https://arxiv.org/pdf/2608.00222",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-31",
  "publisher": "Cornell University",
  "ris": "https://research.draconitic.com/paper/d4332mqram8mr.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2608.00222",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.00222"
    },
    {
      "first_seen": "2026-11-07T13:20:42Z",
      "native_id": "W7172413910",
      "source": "openalex",
      "url": "https://arxiv.org/abs/2608.00222"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Counterexamples to the Jacobian conjecture in dimensions greater than two",
  "title_text": "Counterexamples to the Jacobian conjecture in dimensions greater than two",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:42Z",
  "url": "https://research.draconitic.com/paper/d4332mqram8mr",
  "venue": "arXiv",
  "versions": []
}