Statement copy in the gpt 6 astra pro collection (Math-72)

Two-Dimensional Jacobian Conjecture in Characteristic Zero

Definitions and mathematical statement

For $F=(f,g)$ with $f,g\in k[x,y]$, its Jacobian determinant is \[ JF=\frac{\partial f}{\partial x}\frac{\partial g}{\partial y} -\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}. \] A Keller map satisfies $JF\in k^\times$. In characteristic zero, the question asks whether there are $u,v\in k[x,y]$ making $G=(u,v)$ a two-sided polynomial inverse: $G\circ F=F\circ G=\operatorname{id}$. This is the affine plane case over every characteristic-zero field. Statements or alleged counterexamples in more variables do not decide this selected two-variable target.

Formulation and concept references: [S1].

Short English statement

Does every plane polynomial map with constant nonzero Jacobian determinant have a polynomial inverse?

Sources

• S1 Terence Tao, "A digestion of the Jacobian conjecture counterexample", What's new, 21 July 2026. https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/. Catalogue formulation source.
• S2 Shuhong Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two. https://arxiv.org/abs/2608.00222. Catalogue research/status reference; not a proof endorsement.
• S3 Gao, section 2: Background and previous work. https://arxiv.org/html/2608.00222v1#S2. Catalogue research/status reference; not a proof endorsement.
• E1 J. A. Guccione, J. J. Guccione, R. Horruitiner, and C. Valqui, Increasing the degree of a possible counterexample to the Jacobian Conjecture from 100 to 108, arXiv:2204.14178v1, 29 April 2022. Theorem 2.1 and Sections 2--6; the total-degree exclusion is distinct from this notebook's degree-in-one-variable calculation. https://arxiv.org/html/2204.14178v1. Consulted 25 September 2026; cited as a partial-result manuscript, not as a proof of the full conjecture.