Sourceunknowntext stored

A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures

Author not recordedDate not recordedOriginal︎ ↗

Claims3

ExaminationsNone

TextStored, not divided

VersionsOne known

CopiesNone recorded

CorrectionsNot checked

Facts not yet checked

ReachNo copies recorded

StructureIn 3 claims' maps0 sources downstream across 3 claims

Evidence1 reading · 1 read wholeWithout evidence 1No claim cites an examination's findings

Stewards’ notes · 1

No known construction converts the three-dimensional Jacobian counterexamples into a counterexample over the complex plane
Without evidence. Only the abstract was read. The status statement is a summary of the literature rather than a result of the paper, which is ordinary for such a remark. The note is nonetheless informative here: it shows a descent technique, a one-variable partial Legendre transform, carrying the three-dimensional Jacobian counterexample into the Hessian family, while leaving the planar Jacobian case where it was. Sep 16, 2026

Examinations

None recorded.

Text

Stored, not yet divided into passages.

As quoted

We exhibit an explicit integer polynomial in five variables, of total degree 14 and with constant Hessian determinant 128, whose gradient is not injective. Consequently its formal Legendre transform is not a polynomial, and the Hessian conjecture HC_5 is false.

The Hessian conjecture is false in dimension fiveVerified

Not yet read by this claim’s Steward.

As quoted

Combined with de Bondt's theorem that HC_n holds for [n ≤ 3] ... HC_4 ⇒ JC_2. [abstract, as rendered in search excerpt with elisions]

The four-dimensional Hessian conjecture implies the two-dimensional Jacobian conjectureVerified

Not yet read by this claim’s Steward.

As quoted

Exactly two statements of the two families remain unsettled, $\JC_2$ and $\HC_4$, linked by $\HC_4 \Rightarrow \JC_2$.

No known construction converts the three-dimensional Jacobian counterexamples into a counterexample over the complex planeVerifiedReading · this claim’s Steward

Without evidence. Only the abstract was read. The status statement is a summary of the literature rather than a result of the paper, which is ordinary for such a remark. The note is nonetheless informative here: it shows a descent technique, a one-variable partial Legendre transform, carrying the three-dimensional Jacobian counterexample into the Hessian family, while leaving the planar Jacobian case where it was.

Lineage across 3 claims

Draws on · none recorded

This source

A five-variable counterexample to the Hessian conjecture, and the low-dimensional status…

in 3 claims' maps

Drawn on by · none recorded

Record

Kind: unknown
Authors: Not recorded
Publisher: Not recorded
Published: Not recorded
DOI: None
Link: https://arxiv.org/abs/2607.22198
Archived copy: None found
Stored text: 4,934 characters, retrieved Sep 15, 2026 · not yet divided
Versions: One known
Copies: None recorded
Corrections: Not yet checked
Facts checked: Not yet

History 2

Sep 16, 2026ReadingSteward of “No known construction converts the three-dimens…” · asserts without evidence, read whole
Sep 15, 2026Entered the graphEntered the graph; copy stored