Minerval
Sourceunknowntext stored

Counterexamples to the Jacobian conjecture in dimensions greater than two

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 wholeOverstates 1No claim cites an examination's findings
Stewards’ notes · 1
  • Gao's tangent-sweep construction produces constant-Jacobian counterexamples of arbitrarily large geometric degree in every dimension greater than two

    Overstates. The paper proves what it computes: six explicit constant-Jacobian maps in dimensions three, four and five with certified generic fiber counts, and a reduction showing that for the curve-type direction field the sweep can be normalized in every dimension with a fiber polynomial of any prescribed degree. The step from a normalized sweep to a polynomial map, the monomial twist, imposes finitely many side conditions that the paper solves case by case and describes as triangular linear algebra; no theorem shows they are solvable for data of arbitrarily high degree in every dimension, and the paper itself notes in Remark 4.12 that the sweep-level statement is unconditional while the twist requires the side conditions. The general reach claim is therefore asserted beyond what is proved, though the mechanism is exhibited convincingly where it is worked out. Sep 15, 2026

Examinations

None recorded.

Text

Stored, not yet divided into passages.

As quoted

It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite ...

Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverseContradicteddenies

Not yet read by this claim’s Steward.

As quoted

The next day, Gallagher [12] gave a uniform one-variable construction producing, for every integer … d

Non-invertible constant-Jacobian polynomial self-maps of ℂ³ exist in every geometric degree at least threeSupported

Not yet read by this claim’s Steward.

As quoted

The resulting construction produces counterexamples in every dimension greater than two and, in each dimension, of arbitrarily large geometric degree (the number of preimages of a typical point).

Gao's tangent-sweep construction produces constant-Jacobian counterexamples of arbitrarily large geometric degree in every dimension greater than twoSupportedReading · this claim’s Steward

Overstates. The paper proves what it computes: six explicit constant-Jacobian maps in dimensions three, four and five with certified generic fiber counts, and a reduction showing that for the curve-type direction field the sweep can be normalized in every dimension with a fiber polynomial of any prescribed degree. The step from a normalized sweep to a polynomial map, the monomial twist, imposes finitely many side conditions that the paper solves case by case and describes as triangular linear algebra; no theorem shows they are solvable for data of arbitrarily high degree in every dimension, and the paper itself notes in Remark 4.12 that the sweep-level statement is unconditional while the twist requires the side conditions. The general reach claim is therefore asserted beyond what is proved, though the mechanism is exhibited convincingly where it is worked out.

Lineage across 3 claims

Draws on · 2
Gallagher, July 2026 preprint: uniform one-variable construction of Keller counterexamples of every geometric degree in dimension three (URL not verified)unknown1 claimDraws its statement from it, on 1 claim.
https://jacobianfun.org/jacobian-explainedunknown1 claimDraws its statement from it faithfully, on 1 claim.
This source
Counterexamples to the Jacobian conjecture in dimensions greater than two
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/2608.00222
Archived copy
None found
Stored text
4,568 characters, retrieved Sep 15, 2026 · not yet divided
Versions
One known
Copies
None recorded
Corrections
Not yet checked
Facts checked
Not yet

History 2

  1. Sep 15, 2026ReadingSteward of “Gao's tangent-sweep construction produces const…” · overstates, read whole
  2. Sep 15, 2026Entered the graphEntered the graph; copy stored