HEXAGONbeta

The four-dimensional Hessian conjecture for potentials of degree at most five

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 HC_n asserts that if f in C[x_1,...,x_n] has det Hess(f) in C^×, then ∇f is a polynomial automorphism. It is known for n ≤ 3, false for n ≥ 5, and open for n = 4, where it implies the plane Jacobian conjecture. Ni proved HC_4 for potentials of degree at most four, and the author treated degree five when the top form depends on exactly three linear forms (hexagon:2611.00134). We prove HC_4 for every potential of degree at most five. The remaining top forms are a binary form or a fifth power of a linear form. Both cases are reduced, by weighted layers of the Hessian determinant and exact computer algebra over Q, to four elementary mechanisms: a Schur complement in one variable (which reduces to HC_3), a partial Legendre transform in two variables (which reduces to HC_2), a "dead direction" lemma (which reduces to HC_2 and the plane Jacobian conjecture in degrees at most (3,4), known by Moh), and explicit triangular inverses. Five self-contained verifiers are supplied as ancillary files. The paper has not been refereed.

Provenance statement

The mathematics and the writing were produced with the assistance of an AI model; the author checked the statements and the references.

Tools used

Anthropic
Claude Opus

References

  1. A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc.\ Amer.\ Math.\ Soc.\ 13 (1962), 200--203.
  2. M. de Bondt, Polynomials with constant Hessian determinants in dimension three, J. Pure Appl.\ Algebra 219 (2015), no. 9, 3743--3754.
  3. M. de Bondt, Quasi-translations and singular Hessians, Colloq.\ Math.\ 152 (2018), no. 2, 175--198.
  4. M. de Bondt and A. van den Essen, Singular Hessians, J. Algebra 282 (2004), no. 1, 195--204.
  5. F. Dillen, Polynomials with constant Hessian determinant, J. Pure Appl.\ Algebra 71 (1991), no. 1, 13--18.
  6. Anonymous, Structural reductions toward the quartic Hessian conjecture in dimension four, v0.1.0-candidate, Zenodo, doi:10.5281/zenodo.22216400 (2026).DOI
  7. P. Gordan and M. Nöther, Über die algebraischen Formen, deren Hesse'sche Determinante identisch verschwindet, Math.\ Ann.\ 10 (1876), 547--568.
  8. H. Liu, On the Hessian conjecture in Lorentzian signature: constant pivots and Hesse systems, arXiv:2608.19112 (2026).arXiv
  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 14 referencesShow fewer references
  1. T. T. Moh, On the Jacobian conjecture and the configurations of roots, J. Reine Angew.\ Math.\ 340 (1983), 140--212.
  2. D. Niedbala Giraudin, The four-dimensional Hessian conjecture for quintic potentials with a ternary cone top, Hexagon, hexagon:2611.00134 (2026).hexagon
  3. A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Progr.\ Math.\ 190, Birkhäuser, 2000.
  4. Z. Ni, The quartic Hessian conjecture in dimension four, arXiv:2608.14217 (2026).arXiv

Version history

  1. v1 · 7 pagesSubmitted by David Niedbala GiraudinInitial depositCurrentNov 06, 2026