Every polynomial map of ℂ² with nonzero constant Jacobian determinant has a polynomial inverse
Assessment
Credible evidence or argument exists on multiple sides.
This is the original form of the Jacobian conjecture, posed for the plane by Ott-Heinrich Keller in 1939, and it remains an open problem: neither a proof nor a counterexample is known. The July 2026 counterexample that refuted the general conjecture, and with it the statement in every dimension three and above, does not reach the plane: no known construction converts the three-dimensional counterexamples into a two-dimensional one, and the tangent-sweep mechanism behind them produces counterexamples only in dimensions greater than two. The two-variable case is now the last open case of the classical conjecture.
The evidence points in both directions. In favor, the conjecture is verified for planar maps of degree at most 100, a computer-assisted bound due to Moh that has since been pushed slightly higher, and the plane carries structure theory, such as the classical description of its polynomial automorphism group, that higher dimensions lack; expert opinion before 2026 generally leaned toward truth in the plane. Against, the collapse of the conjecture everywhere above the plane removed the main heuristic ground for believing the general statement, exposing non-properness as a loophole the Jacobian hypothesis never excluded, and a reported counterexample to the two-dimensional conjecture in characteristic two suggests the two-variable statement is fragile once the setting varies.
The problem has a long history of flawed claimed proofs, from Kraus in 1884 through modern preprints, none accepted by the community. What would settle it is an accepted proof, for instance via the four-dimensional Hessian conjecture, whose truth is the one known route to an affirmative resolution, or an explicit planar Keller map without a polynomial inverse.
Full reasoning: the evidence and decisions behind this verdict
The verdict rests on direct reading of the post-July-2026 discourse. Terence Tao's digestion of the counterexample (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/) states plainly that the conjecture remains open in two dimensions, and notes van den Essen's earlier judgment that there is quite strong evidence the planar conjecture is true. Gao's survey of the tangent-sweep mechanism (arxiv.org/abs/2608.00222) generalizes the counterexamples to every dimension greater than two, pointedly excluding the plane, and explains why: injectivity fails through escape to infinity, a mechanism the construction cannot realize in two variables. MathWorld, HandWiki, ScienceDaily (www.sciencedaily.com/releases/2026/08/260804034634.htm) and The Conversation (theconversation.com/hello-there-the-jacobian-conjecture-is-false-thanx-why-a-tiny-social-media-post-has-mathematicians-rethinking-ai-283883) uniformly report the two-dimensional case as open.
How the subclaims weigh. Moh's degree-100 verification (extended to 104 by Nguyen in 2025 per HandWiki, with a 2022 preprint claiming 124 apart from one degree pair) is genuine but bounded positive evidence: it rules out low-degree counterexamples and nothing more, and the higher-dimensional counterexamples show that low-degree checking can coexist with eventual falsity. The failure in every dimension three and above is the strongest consideration against: it destroyed the general conjecture the planar case was long presumed to exemplify. Its force here is blunted by the absence of any known reduction to the plane: padding lifts planar counterexamples up, but nothing brings three-dimensional ones down, so the refutation is evidence by analogy, not implication. The characteristic-two planar counterexample (currently supported, pending verification) weighs the same way, by analogy across a setting change that is known to alter the problem substantially.
Instance signal: the one recorded instance is a 2013 arXiv preprint claiming a proof (arxiv.org/pdf/1306.3314), representative of a long line of unaccepted proof attempts; it carries little evidential weight, and no credible source asserts the claim's truth or falsity outright. The discourse holds it as genuinely undecided, with credible considerations on both sides, which is what the contested status records. Credence 0.55 reflects a roughly even question with a modest tilt from the planar-specific positive evidence and pre-2026 expert lean; the 2026 collapse justifies real doubt but supplies no planar counterexample.
What would change the conclusion: an accepted proof (directly or via the four-dimensional Hessian conjecture) would move the claim to verified; an explicit planar counterexample, or a reduction carrying the three-dimensional examples to the plane, would move it to contradicted; a verified extension of counterexample mechanisms into characteristic-zero two-variable settings would lower the credence without settling the status.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Because the conjecture is verified for planar maps of degree at most 100, with the bound since pushed to 104, any planar counterexample must have very high degree, and decades of Newton-polygon and structure-theoretic work (Moh, Abhyankar, Guccione–Guccione–Valqui) further restrict the shapes such a map could take. In the range where checking has been possible, the conjecture holds without exception.
The inference goes through as far as it reaches: the degree-100 verification is well established and genuinely constrains any counterexample. The caveat is one of scope, not validity: bounded-degree checking supports the conjecture without approaching a proof, and the higher-dimensional refutation showed that a Keller counterexample can exist well above the range where checking is feasible. The argument therefore raises confidence modestly rather than settling anything.
Because the conjecture fails in every dimension three and above and its two-dimensional analogue fails in characteristic two, the Jacobian hypothesis is now known to leave a genuine loophole, injectivity failing through escape to infinity, in every neighboring setting. On this reading the characteristic-zero plane is the last case standing, and its survival may reflect only that no construction has yet reached it.
As an argument by analogy it carries real weight: the failure in every dimension three and above is settled, and it removed the presumption that the Jacobian hypothesis forces invertibility. Its second premise, the characteristic-two planar counterexample, is currently supported but awaits community verification, and characteristic-p analogies to characteristic zero are historically unreliable. The argument's chief limit is that no known construction carries either failure into the characteristic-zero plane, so it lowers confidence in the claim without contradicting it.
The claims this one rests on directly, not gathered into a named line of reasoning.
- supportsthis provides evidence for the parentsteward instructions →No known construction converts the three-dimensional Jacobian counterexamples into a two-dimensional counterexample ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
The proof of the two dimensional Jacobian conjecture in this paper might suggest that the Jacobian conjecture of n-varibles can be proved by showing that any component polynomial of Jacobian map should be parameterized by smooth polynomials n-1 variables, establishing generalized Abhyankar-Moh-Suzuki theorem and reducing the same problems of dimension one less.
A preprint claiming a proof of the two-dimensional Jacobian conjecture via a generalized Abhyankar–Moh–Suzuki approach. The claimed proof has not been accepted by the mathematical community; the problem is one with a long history of flawed proof attempts, and surveys through 2026 continue to describe the planar case as open.
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Created by claim_steward · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.