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Jacobian Conjecture

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The Jacobian conjecture asserts that a polynomial map F:C^n->C^n having a nonzero constant Jacobian determinant is an automorphism. In the plane, first stated by Keller (1939), it says that a ring map F of C[x,y] (the polynomial ring in two variables over the complex numbers) to itself that fixes C and sends x, y to f, g, respectively, is an automorphism iff the Jacobian f_xg_y-f_yg_x 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).

JacobianConjectureCounterexample

In July 2026, Alpöge (2026) announced the following polynomial counterexample, which he credited to the AI system Fable. Writing F=(P,Q,R), its coordinate polynomials are

P(x,y,z)=(1+xy)^3z+y^2(1+xy)(4+3xy)
(1)
Q(x,y,z)=y+3x(1+xy)^2z+3xy^2(4+3xy)
(2)
R(x,y,z)=2x-3x^2y-x^3z.
(3)

Its Jacobian determinant is the nonzero constant

detJ_F=-2,
(4)

but the three distinct points

(0,0,-1/4),(1,-3/2,(13)/2),(-1,3/2,(13)/2),
(5)

have the same image, since

F(0,0,-1/4)=F(1,-3/2,(13)/2)=F(-1,3/2,(13)/2)=(-1/4,0,0).
(6)

Therefore, the Jacobian conjecture is false in dimension 3 and, by adjoining identity coordinates, in every dimension n>=3, 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 Jacobian conjecture is one of Smale's problems.

See also

Image Conjecture, Invertible Polynomial Map, Jacobian, Polynomial Map, Smale's Problems, Vanishing Conjecture

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References

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.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. Monthly 102, 546-550, 1995.Smale, S. "Mathematical Problems for the Next Century." Math. Intelligencer 20, 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.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/.

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Jacobian Conjecture

Cite this as:

Weisstein, Eric W. "Jacobian Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JacobianConjecture.html

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