On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems

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(Hess(ϕ))=±1 is also a polynomial solution. The general Hessian conjecture is false for n≥5, 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_ξ²ϕ 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+ϕ₂+ϕ₁+ϕ₀ with d≥4. More generally, we prove the same conclusion when ϕ=ϕ_d+⋯+ϕ_d−k+ϕ₂+ϕ₁+ϕ₀, where k≥0 and d≥4k+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 rank(μ_ϕ)≤55. 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(Hess(ϕ−ϕ₂))≡0, then the potential ϕ admits a pivot. After that, we then give an analytic degeneracy criterion for Hess(ϕ−ϕ₂). Finally, we prove the Hessian conjecture in this setting for every polynomial potential of degree at most five.

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BibTeX RIS

BibTeX

@misc{liu2026hessian,
  title = {{On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems}},
  author = {Hanwen Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.19112},
  eprint = {2608.19112},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.19112},
}

RIS

TY  - UNPB
TI  - On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems
AU  - Hanwen Liu
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2608.19112
UR  - https://arxiv.org/abs/2608.19112
AB  - 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(Hess(ϕ))=±1 is also a polynomial solution. The general Hessian conjecture is false for n≥5, 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≥4. More generally, we prove the same conclusion when ϕ=ϕ_d+⋯+ϕ_d−k+ϕ_2+ϕ_1+ϕ_0, where k≥0 and d≥4k+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 rank(μ_ϕ)≤55. 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(Hess(ϕ−ϕ_2))≡0, then the potential ϕ admits a pivot. After that, we then give an analytic degeneracy criterion for Hess(ϕ−ϕ_2). Finally, we prove the Hessian conjecture in this setting for every polynomial potential of degree at most five.
ER  -

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