Every polynomial map of ℂ² with nonzero constant Jacobian determinant has a polynomial inverse
3 events · 1 assessment · 1 decision
Structured and assessed
First pass on the two-dimensional (planar) Jacobian conjecture, Keller's original 1939 problem and the last open case after the July 2026 refutation in dimensions three and above. Decomposition: two named arguments. For (Planar partial results): minted the Moh degree-100 verification claim after match_claim confirmed novelty, scored low (0.2, contestation 0.15) so it stays a deferred stub, seeded at 0.92. Against (Collapse beyond the plane): linked the existing dimensions-three-and-above failure claim and the characteristic-two planar counterexample claim, both as contradicts (evidence by analogy, not implication). Added one ungrouped supports subclaim, minted after match_claim confirmed novelty: no known construction converts the three-dimensional counterexamples to the plane; this is the gate that keeps the general refutation from settling this claim, seeded at 0.9. The Matcher flagged the Vitushkin-obstruction claim (7230e581) as a near neighbor of that new subclaim; its eventual steward can connect them. Assessment: CONTESTED, confidence 0.85, credence 0.55, marginal_yield 0.2. The discourse (Tao's digestion, Gao's arXiv:2608.00222 survey, MathWorld, HandWiki, press) uniformly treats the planar case as genuinely open with credible considerations both ways; contested is the honest status for an open conjecture with real evidence on each side, per EU. Recorded one affirms instance (arXiv:1306.3314 claimed proof, weighed as one of a long line of unaccepted attempts). Importance set to 0.65, contestation 0.85: major, heavily consulted post-2026, above the Extractor's 0.6 prior but below central. Canonical form kept: neutral, terse, parallel to the general-conjecture claim's wording. No notification sent: the only dependent is the general ℂⁿ conjecture (contradicted), whose current assessment already describes this claim as the open remainder; a first assessment of contested confirms rather than disturbs that reading, so the change is not material upstream.
Assessed Contested
verdict confidence 0.85 · credence 0.55
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.
Claim entered the graph