{
  "abstract": "As a close relative of the Jacobian conjecture, the Hessian conjecture in dimension $n$ states that the local Legendre transform of a polynomial solution to the Monge--Ampère equation $\\det(\\operatorname{Hess}(ϕ))=\\pm1$ is also a polynomial solution. The general Hessian conjecture is false for $n\\geq5$, while in Riemannian signature it follows from the Jörgens--Calabi--Pogorelov theorem. We study the four-dimensional Hessian conjecture in Lorentzian signature. For a polynomial potential $ϕ$ in four real variables whose Hessian matrix has index $1$ and determinant $-1$, we define a constant pivot for $ϕ$ to be a nonzero constant vector $ξ$ such that the second directional derivative $D_ξ^2ϕ$ is constant. We then prove that the gradient mapping of every potential admitting a pivot is a polynomial automorphism, and that a pivot always exists when $ϕ$ decomposes into homogeneous pieces as $ϕ=ϕ_d+ϕ_{d-1}+ϕ_2+ϕ_1+ϕ_0$ with $d\\geq4$. More generally, we prove the same conclusion when $$ϕ=ϕ_d+\\cdots+ϕ_{d-k}+ϕ_2+ϕ_1+ϕ_0,$$ where $k\\geq0$ and $d\\geq4k+3$. Then, we associate with each potential a linear system of quadrics, called the Hesse system, and a canonical homomorphism $μ_ϕ$. We prove that the existence of a pivot is equivalent to $\\operatorname{rank}(μ_ϕ)\\leq55$. As an application, we prove that the gradient mapping is a polynomial automorphism whenever the Hesse system has complex dimension at most 4. We also show that if ${\\det(\\operatorname{Hess}(ϕ-ϕ_2))\\equiv0}$, then the potential $ϕ$ admits a pivot. After that, we then give an analytic degeneracy criterion for $\\operatorname{Hess}(ϕ-ϕ_2)$. Finally, we prove the Hessian conjecture in this setting for every polynomial potential of degree at most five.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19112",
  "author_total": 1,
  "authors": [
    {
      "name": "Hanwen Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b7x5zmgd4k79a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19112",
  "fulltext_url": null,
  "id": "b7x5zmgd4k79a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19112",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7203886500",
  "pdf_url": "https://arxiv.org/pdf/2608.19112",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b7x5zmgd4k79a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2608.19112",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19112"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems",
  "title_text": "On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/b7x5zmgd4k79a",
  "venue": "arXiv",
  "versions": []
}