HEXAGONbeta

The four-dimensional Hessian conjecture for quintic potentials with a ternary cone top

Primarily AI-generated textHuman understanding: no partsmath.AG — Algebraic Geometrymath.AC — Commutative Algebra

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

The Hessian conjecture HCn\mathrm{HC}_n asserts that if f∈C[x1,…,xn]f\in\mathbb{C}[x_1,\dots,x_n] has det⁡Hess⁡(f)∈C×\det\operatorname{Hess}(f)\in\mathbb{C}^\times, then ∇f\nabla f is a polynomial automorphism. It is known for n≤3n\le 3, false for n≥5n\ge 5, and open for n=4n=4, where it implies the plane Jacobian conjecture. Z. Ni proved HC4\mathrm{HC}_4 for potentials of degree at most four. We prove HC4\mathrm{HC}_4 for every potential of degree five whose top homogeneous form depends on exactly three linear forms. The proof combines a weighted analysis of the Hessian determinant with exact computer-algebra certificates (Groebner bases and Rabinowitsch tests over Q\mathbb{Q}); one branch reduces to plane Jacobian pairs of degrees (4,3)(4,3), which are excluded by Magnus' theorem. A self-contained verifier is supplied as an ancillary file. The binary and unary cone tops in degree five remain open.

Author comments: 5 pages. Ancillary file anc/verify_hc4_ternary.py: 29 exact checks (python-flint, sympy, Singular via Sage), about 3 minutes. Extends arXiv:2608.14217 to degree five for ternary cone tops. Computations and writing assisted by Claude (Anthropic).

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by an independent AI review and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

  1. M. de Bondt, Polynomials with constant Hessian determinants in dimension three, J. Pure Appl.\ Algebra 219 (2015), no. 9, 3743--3754.
  2. M. de Bondt, Quasi-translations and singular Hessians, Colloq.\ Math.\ 152 (2018), no. 2, 175--198.
  3. M. de Bondt and A. van den Essen, Singular Hessians, J. Algebra 282 (2004), no. 1, 195--204.
  4. F. Dillen, Polynomials with constant Hessian determinant, J. Pure Appl.\ Algebra 71 (1991), no. 1, 13--18.
  5. Anonymous, Structural reductions toward the quartic Hessian conjecture in dimension four, v0.1.0-candidate, Zenodo, doi:10.5281/zenodo.22216400 (2026).DOI
  6. A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Progr.\ Math.\ 190, Birkhäuser, 2000.
  7. P. Gordan and M. Nöther, Über die algebraischen Formen, deren Hesse'sche Determinante identisch verschwindet, Math.\ Ann.\ 10 (1876), 547--568.
  8. A. Magnus, On polynomial solutions of a differential equation, Math.\ Scand.\ 3 (1955), 255--260.
  9. G. Meng, Legendre transform, Hessian conjecture and tree formula, Appl.\ Math.\ Lett.\ 19 (2006), no. 6, 503--510.
  10. G. Meng and L. Yang, A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures, arXiv:2607.22198 (2026).arXiv
Show all 11 referencesShow fewer references
  1. Z. Ni, The quartic Hessian conjecture in dimension four, arXiv:2608.14217 (2026).arXiv

Version history

  1. v1 · 5 pagesSubmitted by David Niedbala Giraudin5 pages. Ancillary file anc/verify_hc4_ternary.py: 29 exact checks (python-flint, sympy, Singular via Sage), about 3 minutes. Extends arXiv:2608.14217 to degree five for ternary cone tops. Computations and writing assisted by Claude (Anthropic).CurrentNov 05, 2026