The Jacobian conjecture asserts that a polynomial map having a nonzero
constant Jacobiandeterminant
is an automorphism. The plane statement appeared
as the purported main result of a flawed proof by Kraus (1884), more than half a
century before Keller (1939) posed the problem (Rodríguez Díaz 2026).
Equivalently, in two dimensions it asserts that a ring
homomorphism
from
(the polynomial ring in two variables over the complex
numbers) to itself that fixes and sends , to , , respectively, is an automorphismiff the Jacobian is a nonzero constant. The condition is easily
shown to be necessary.
There have been at least five published incorrect proofs and many incorrect attempts over the years (cf. Hochster 2004, Woit 2004).
In July 2026, Alpöge (2026) announced the following polynomial counterexample, which he credited to Anthropic's Claude Fable 5 AI model (Stark-Elster 2026). Writing
,
its coordinate polynomials are
Therefore, the Jacobian conjecture is false in dimension 3 and, by adjoining identity coordinates, in every dimension , while the plane case remains open (Zhang 2026).
Known all-dimensional implications then show that the image conjecture and vanishing conjecture are
also false in some finite dimension (Zhang 2026).
The closely related dimension-Hessian conjecture asks
whether the formal Legendre transformation
of a polynomial with nonzero constant Hessiandeterminant is a polynomial.
An explicit five-dimensional counterexample, together with low-dimensional results,
shows that it is true for , false for , and open only for (Meng and Yang 2026). Thus the only unresolved cases in
the two indexed families are and . If the Hessian conjecture is true in dimension 4, then
the Jacobian conjecture is true in dimension 2.
Abhyankar, S. S. Lectures on Expansion Techniques in Algebraic Geometry. Bombay, India: Tata Institute
of Fundamental Research, 1977.Alpöge, L. X post, July 20, 2026.
https://x.com/__alpoge__/status/2079028340955197566.Bass,
H. "Conjecture jacobienne et opérateurs différentiels." Mém.
Soc. Math. France, No. 38, 39-50, 1989.Bass, H.; Connell, E. H.;
and Wright, D. "The Jacobian Conjecture: Reduction of Degree and Formal Expansion
of the Inverse." Bull. Amer. Math. Soc.7, 287-330, 1982.Becker,
T. and Weispfenning, V. Gröbner
Bases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag,
p. 330, 1993.Drużkowski, L. M. "The Jacobian Conjecture."
IMPAN Preprint 492. Kraków, Poland: Math. Inst. Jagiellonian University, 1991.Formanek,
E. "Observations about the Jacobian Conjecture." Houston J. Math.20,
369-380, 1994.Hochster, M. "The Jacobian Conjecture in the Plane
Is Solved." Email of Nov. 5, 2004, forwarded by I. Agol to sci.math.research,
Nov. 10, 2004. https://groups.google.com/g/sci.math.research/c/zeApl3rDV5o.Keller,
O.-H. "Ganze Cremona Transformationen." Monatsh. für Math. u. Phys.47,
299-306, 1939.Kraus, L. "Über Functionaldeterminanten."
Wien. Ber.90, 813-826, 1884. https://archive.org/details/sitzungsbericht398klasgoog/page/820.Meisters,
G. H. "Jacobian Problems in Differential Equations and Algebraic Geometry."
Rocky Mountain J. Math.12, 679-705, 1982.Meisters, G. H.
"Wanted: A Bad Matrix." Amer. Math. Monthly102, 546-550,
1995.Meng, G. and Yang, L. "A Five-Variable Counterexample to the
Hessian Conjecture, and the Low-Dimensional Status of the Jacobian and Hessian Conjectures."
July 27, 2026. https://arxiv.org/abs/2607.22198.Rodríguez
Díaz, L. O. "On the Origin of the Jacobian Conjecture." C. R.
Math.364, 363-370, 2026. https://doi.org/10.5802/crmath.831.Smale,
S. "Mathematical Problems for the Next Century." Math. Intelligencer20,
No. 2, 7-15, 1998.Smale, S. "Mathematical Problems for the
Next Century." In Mathematics:
Frontiers and Perspectives 2000 (Ed. V. Arnold, M. Atiyah, P. Lax,
and B. Mazur). Providence, RI: Amer. Math. Soc., 2000.Stark-Elster,
E. "A.I. Disproves a Decades-Old Mathematical Idea, the 'Biggest Conjecture'
That the Tech Has Played a Role in Yet." Smithsonian Magazine, July 27,
2026. https://www.smithsonianmag.com/smart-news/ai-disproves-a-decades-old-mathematical-idea-the-biggest-conjecture-that-the-tech-has-played-a-role-in-yet-180989189/.Woit,
P. "Not Even Wrong: Proof of the Jacobian Conjecture." Nov. 10, 2004.
https://www.math.columbia.edu/~woit/wordpress/?p=105.Zhang,
Z. "Direct Consequences of the Three-Dimensional Counterexample to the Jacobian
Conjecture." July 20, 2026. https://zzhang-iu.github.io/papers/direct-consequences-jacobian/.
having a nonzero
constant Jacobian determinant
is an automorphism. The plane statement appeared
as the purported main result of a flawed proof by Kraus (1884), more than half a
century before Keller (1939) posed the problem (Rodríguez Díaz 2026).
Equivalently, in two dimensions it asserts that a ring
homomorphism 
from ![C[x,y]](/images/equations/JacobianConjecture/Inline3.svg)
(the polynomial ring in two variables over the complex
numbers) to itself that fixes 
and sends 
, 
to 
, 
, respectively, is an automorphism iff the Jacobian 
is a nonzero constant. The condition is easily
shown to be necessary.
,
its coordinate polynomials are
is the nonzero constant
, while the plane case remains open (Zhang 2026).
Hessian conjecture asks
whether the formal Legendre transformation
of a polynomial with nonzero constant Hessian determinant is a polynomial.
An explicit five-dimensional counterexample, together with low-dimensional results,
shows that it is true for 
, false for 
, and open only for 
(Meng and Yang 2026). Thus the only unresolved cases in
the two indexed families are 
and 
. If the Hessian conjecture is true in dimension 4, then
the Jacobian conjecture is true in dimension 2.
